4.0. Introductionneutrino.aquaphoenix.com/ReactionDiffusion/chap4.pdf · 122 4. APPLICATIONS IN...

CHAPTER 4

APPLICATIONS IN ASTROPHYSICS AND STOCHASTICPROCESSES

[The physics part of this chapter is based on the lectures of Professor D r. H ans Hauboldof the Outer Space D ivision of the United Nations and the stochastic process part is based onthe lectures of D r. R .N . P illai, D r. K .K . Jose, D r. V. Seethalekshm i, D r. A lice Thomas, D r. K .Jayakumar, D r. E . Sandhya and D r. S . Satheesh.]

4.0. Introduction

Since the participants of the Third S.E.R.C. school are of different backgroundsit is not worth discussing the individual topics in astrophysics and stochastic pro-cesses. Only the mathematical aspects where the various special functions come innaturally will be discussed here. We start with a few topics in astrophysics suchas solar models, energy generations in stars, gravitational instability problems andreaction-diffusion problems. As stochastic process problems, we will only deal withMittag-Leffler process and the associated applications.

4.1. Solar Models

When looking at the internal structure of the Sun one has to look at mass con-servation, hydrostatic equilibrium, energy conservationand energy transport. Thesecan be described by a system of non-linear differential equations which cannot befully solved analytically. Hence the standard techniques adopted are numerical eval-uations thereby resulting in computer-generated solar models. In order to deriveanalytic models a starting point would be to look into the density distribution in thecore of the Sun. It is well known that the matter density decreases from the cen-ter to the exterior, temperature and pressure also behave the same way. The solarcore is a more stabilized region and hence an analytic model for the matter density

121

122 4. APPLICATIONS IN ASTROPHYSICS AND STOCHASTIC PROCESSES

distribution in the solar core is an appropriate starting point. Let r be an arbitrarydistance from the center of the Sun andR⊙ the solar radius. The simplest model formatter densityρ(r) is a linear model of the type

ρ(r) = ρ0 [1 −r

R⊙] ( Model 1 )

which indicates that the matter density decreases linearlyfrom the core to the sur-face and it is zero at the surface and it is a constantρ0 at the center. But observationsindicate that a linear model is not correct. A more appropriate starting point wouldbe to consider the non-linear model

ρ(r) = ρ0

[

1−(

rR⊙

)δ]

, δ > 0 ( Model 2 ) (4.1.1)

whereδ is an arbitrary parameter. Since the surface area of a sphereof radiusr is4πr2 the mass of the Sun can be computed from the relation.

ddr

M(r) = 4πr2ρ(r). (4.1.2)

That is,

M(r) = 4πρ0

∫ r

0t2[

1−(

tR⊙

)δ]

dt

=4π3ρ0r

3[

1− 3(δ + 3)

(

rR⊙

)δ]

. (4.1.3)

Denoting the total mass byM⊙, this gives the central density

ρ0 =34π

(δ + 3)δ

M⊙R⊙

. (4.1.4)

From the connection between pressureP(r), massM(r) and densityρ(r), namely,

ddr

P(r) = GM(r)ρ(r)

r2(4.1.5)

whereG is the gravitational constant, we have

4.1. SOLAR MODELS 123

P(r) = P(0)−G∫ r

0

M(t)ρ(t)t2

dt

=4πG

3ρ2

0R2⊙

{

ξ − 12

(

rR⊙

)2

+(δ + 6)

(δ + 2)(δ + 3)

(

rR⊙

)δ+2

−3

2(δ + 1)(δ + 3)

(

rR⊙

)2δ+2}

(4.1.6)

where

ξ =12−

(δ + 6)(δ + 2)(δ + 3)

+3

2(δ + 1)(δ + 3). (4.1.7)

From the relationship between temperatureT(r) and pressureP(r), namely,

T(r) =µ

kNA

P(r)ρ(r)

(4.1.8)

whereµ is the mean molecular weight,NA is Avogadro’s constant andk is theBoltzmann constant, we have

T(r) =4πG µ ρ0

3kNA

R2⊙

[1 − ( rR⊙

)δ]

{

ξ − 12

( rR⊙

)2

+(δ + 6)

(δ + 2)(δ + 3)

( rR⊙

)δ+2

−3

2(δ + 1)(δ + 3)

( rR⊙

)2δ+2}

. (4.1.9)

From the equation of energy conservation, namely,

ddr

L(r) = 4πr2ρ(r)ǫ(r) (4.1.10)

whereL(r) represents the energy flux through the sphere with radiusr so thatL(R⊙)represents the luminosity of the Sun andǫ(r) is the rate of thermonuclear energygeneration per unit mass including the tiny energy losses via solar neutrinos, we cancompute luminosity once an expression is available forǫ(r).

If we consider a specific reaction, say particles 1 and 2 reacting to give rise toparticles 3 and 4 or 1+ 2→ 3+ 4 then the internal luminosity due to this specificreaction is given by


L12(R⊙) =∫ R⊙

04πr2ρ(r)ǫ12(r) dr (4.1.11)

A general model that can be used to representǫ(r) is the following:

ǫ(r) = ǫ0

[

ρ(r)ρ0

]α [

T(r)T0

]β

, (4.1.12)

whereα andβ are real constants.

4.1.1. A more general model for density

A more general two-parameter model for the matter density distribution is thefollowing:

ρ(r) = ρ0

[

1−( rR⊙

)δ]γ

, δ > 0, γ > 0 (Model 3) (4.1.13)

Example 4.1.1. Evaluate the total massM(R⊙) as well as the mass contained in a sphereof arbitrary radiusr under the model in (4.1.13).

Solution: The total mass is given by

M⊙ =∫ R⊙

04πr2 ρ0

[

1−( rR⊙

)δ]γ

dr (4.1.14)

= 4πR3⊙ ρ0

∫ 1

0x2

[

1− xδ]γ

dx, x =r

R⊙

= 4πR3⊙ ρ0

∫ 1

0y

3δ−1 [

1− y]γ dy, y = xδ

= 4πR3⊙ ρ0

Γ

(

3δ

)

Γ(γ + 1)

Γ(γ + 1+ 3δ), (4.1.15)

evaluating (4.1.14) with the help of a type-1 beta integral.Mass at an arbitraryradius r is given by

4.1. SOLAR MODELS 125

M(r) =∫ r

04πt2 ρ0

[

1−( tR⊙

)δ]γ

dt

= 4πρ0 R3⊙

∫ r/R⊙

0u2

[

1− uδ]γ

du, u =t

R⊙, 0 ≤ u ≤ 1.

Expanding (1− uδ)γ with the help of a binomial expansion and then integrating outwe have the following:

(1− uδ)γ = (1− uδ)−(−γ)=

∞∑

k=0

(−γ)k

k!(uδ)k.

M(r) = 4πρ0R3⊙

∞∑

k=0

(−γ)k

k!

∫ r/R⊙

0u2+δk du

= 4πρ0R3⊙

∞∑

k=0

(−γ)k

k!(r/R⊙)3+δk

3+ δk

=4πρ0

δ

∞∑

k=0

(−γ)k

k![(r/R⊙)δ]k

k+ 3δ

.

But

1

k+ 3δ

=13δ

(3δ)k

(3δ+ 1)k

.

Hence

M(r) =4πρ0

3 2F1(−γ,3δ

;3δ+ 1; (r/R⊙)

δ). (4.1.16)

Note that the models in (4.1.13) and (4.1.1) are really validonly in the interiorcore of the Sun. The convective zone has entirely a different behavior for the densitydistribution. But for computing the total mass, pressure, temperature and luminositywe have integrated out over the entire length of the solar radius R⊙. This is notappropriate. The integration should have been done only in the interior core of theSun. Hence a more appropriate model is of the following form:


ρ(r) = ρ0 [1 − a(r

R⊙)δ]γ, δ > 0, γ > 0, a > 0 (Model 4). (4.1.17)

Since 1− axδ > 0, x = rR⊙, we have 0≤ x ≤ 1

a1δ

. Hence for the total integral the

range should have been 0≤ r ≤ R⊙

a1δ

Exercises 4.1.

4.1.1. Evaluateρ0 in terms of the total massM⊙ for the Models 1,2,3 and 4.

4.1.2. EvaluateM(r) for the Models 4 in (4.1.17).

4.1.3. Evaluate the expression for pressureP(r) under Models 3 forδ = 2.

4.1.4. Evaluate the expression for temperature under Models 3 forδ = 2.

4.1.5. Evaluate the expression forM(r) under Models 4.

4.2. Solar Thermonuclear Energy Generation

In reaction rate theory when two particles 1 and 2 reacting togive rise to parti-cles 3 and 4, namely 1+2→ 3+4, the basic assumption is that the distribution of therelative velocities of the reacting particles always remains Maxwell-Boltzmannian.Then the distribution of the relative velocities of the particles can be written as

f (v)dv =(

µ

2πkT

) 32

exp{

−µv2

2kT

}

4πv2 dv (4.2.1)

such that∫ ∞

0f (v)dv = 1. In terms of energyE = µv2

2 we have the density forE givenby,

f (E)dE =2

π12 (kT)

32

exp{

− EkT

}

E12 dE. (4.2.2)

If the relative velocity of the interacting particles 1 and 2is v then the thermally aver-aged product of the cross sectionσ, denoted by the expected value ofσv or< σv >

4.2. SOLAR THERMONUCLEAR ENERGY GENERATION 127

has the following expression under Maxwell-Boltzmann velocity distribution (seeMathai and Haubold (1998) ) and in the charged particle case.

< σv >=

( 8πµ

) 12

2∑

ν=0

S(ν)(0)ν!

1

(kT)−ν+12

∫ ∞

0yνe−y−zy−

12 dy. (4.2.3)

The reaction probability integral coming from< σv >, denoted byNν(z), is thengiven by

Nν(z) =∫ ∞

0yνe−y−zy−

12 dy. (4.2.4)

A more general integral, where (4.2.4) is a particular case,is already evaluated inExample 1.6.3 of Chapter 1. From there we note that

Nν(z) =1

π12

G3,00,3[

z2

4|0, 12 ,1+ν], (4.2.5)

whereG(·) is a Meijer’s G-function. Equation (4.2.3) is the situation under non-resonant reactions. But with depleted Maxwell-Boltzmann distribution the reactionprobability integral changes to the following:

Nν(z; δ) =∫ ∞

0yνe−y−zy−

12 e−y

δ

dy. (4.2.6)

With modified Maxwell-Boltzmann distribution the reactionprobability integral hasthe following form:

Nν(z, d) =∫ d

0yνe−y−zy−

12 dy, d < ∞. (4.2.7)

In this case it is a fractional integral. In the resonant situation it can be shown (seeMathai and Haubold (1988) ) that the reaction probability integral has the followingform:

N(a, b, q, g) =∫ ∞

0

e−ay−qy−12

(b− y)2 + g2dy. (4.2.8)

In the corresponding depleted case the above integral will be modified to the fol-lowing:

N(a, b, q, g, δ) =∫ ∞

0

e−ay−qy−12−cyδ

(b− y)2 + g2dy. (4.2.9)


In the corresponding modified resonant case the integral will have the fractionalform

Nd(a, b, q, g, δ) =∫ d

0

e−ay−qy−12−cyδ

(b− y)2 + g2dy. (4.2.10)

All these various cases and the related situations are considered in a series of papersby Haubold and Mathai, some of the earlier ones are availablefrom Mathai andHaubold (1988). Some of the recent works from 1988 to 2005 will be mentionedin the lectures to be given by Professor Dr. Hans Haubold at the Third S.E.R.C.School.

ReferencesChandrasekhar, S. (1967).An Introduction to the Study of Stellar Structure, DoverPublications, New York.

Mathai, A.M. and Haubold, H.J. (1988).Modern Problems in Nuclear and NeutrinoAstrophysics, Akademie-Verlag, Berlin.

4.3. Mittag-Leffler Distribution and Processes

4.3.1. Preliminary concepts

In this section we discuss some preliminary concepts which will be of frequentuse in this chapter.

Definition 4.3.1. Infinite divisibility.A random variablex is said to be infinitely divisible if for everyn ∈ �, there

exists independently and identically distributed random variablesy1n, y2n, . . . , ynn

such thatxd= y1n+y2n+ · · ·+ynn, where

d= denotes equality in distributions. In terms

of distribution functions, a distribution functionF is said to be infinitely divisibleif for every positive integern, there exists a distribution functionFn such thatF =Fn ⋆ Fn ⋆ · · · ⋆ Fn︸︷︷︸

n times

, where⋆ denotes convolution.

This is equivalent to the existence of a characteristic function ϕn(t) for everyn ∈ �such thatϕ(t) = [ϕn(t)]n whereϕ(t) is the characteristic function ofx.

4.3. MITTAG-LEFFLER DISTRIBUTION AND PROCESSES 129

Infinitely divisible distributions occur in various contexts in the modelling of manyreal phenomena. For instance when modelling the amount of rain x that falls in aperiod of lengthT, one can dividex into more general independent parts from thesame family. That is,

xd= xt1 + xt2−t1 + · · · + xT−tn−1.

Similarly, the amount of moneyx paid by an insurance company during a year mustbe expressible as the sum of the corresponding amountsx1, x2, . . . , x52 in each week,That is,

xd= x1 + x2 + · · · + x52.

A large number of distributions such as normal, exponential, Weibull, gamma,Cauchy, Laplace, logistic, lognormal, Pareto, geometric,Poisson, etc., are infinitelydivisible. Various properties and applications of infinitely divisible distributions canbe found in Laha and Rohatgi (1979) and Steutel (1979).

4.3.2. Geometric infinite divisibility

The concept of geometric infinite divisibility (g.i.d.) wasintroduced by Kle-banovet al. (1984). A random variabley is said to be g.i.d. if for everyp ∈ (0, 1),there exists a sequence of independently and identically distributed random vari-ablesx(p)

1 , x(p)2 , . . . such that

yd=

N(p)∑

j=1

x(p)j (4.3.1)

and

P{N(p) = k} = p(1− p)k−1, k = 1, 2, · · ·wherey,N(p) andx(p)

j , j = 1, 2, . . . are independent. The relation (4.3.1) is equiva-lent to

ϕ(t) =∞∑

j=1

[g(t)] j p(1− p) j−1

=pg(t)

1− (1− p)g(t)


whereϕ(t) andg(t) are the characteristic functions ofy andx(p)j respectively.

The class of g.i.d. distributions is a proper subclass of infinitely divisible dis-tributions. The g.i.d. distributions play the same role in ‘geometric summation’as infinitely divisible distributions play in the usual summation of independent ran-dom variables. Klebanovet al. (1984) established that a distribution functionF

with characteristic functionϕ(t) is g.i.d. if and only if exp

{

1−1ϕ(t)

}

is infinitely

divisible. Exponential and Laplace distributions are obvious examples of g.i.d. dis-tributions. Pillai (1990b), Mohanet al. (1993) discuss properties of g.i.d. dis-tributions. It may be noted that normal distribution is not geometrically infinitelydivisible.

4.3.3. Bernstein functions

A C∞–function f from (0,∞) toR is said to be completely monotone if (−1)ndn fdxn≥

0 for all integersn ≥ 0.

A C∞–function f from (0,∞) to R is said to be a Bernstein function, iff (x) ≥ 0,

x > 0 and (−1)ndn fdxn≤ 0 for all integersn ≥ 1. Then f is also referred to as a

function with complete monotone derivative (c.m.d).

A completely monotone function is positive, decreasing andconvex while aBernstein function is positive, increasing and concave (see Berg and Forst (1975)).

Fujita (1993) established that a cumulative distribution functionG with G(0) =0 is geometrically infinitely divisible, if and only ifG can be expressed as

G(x) =∞∑

n=1

(−1)n+1Wn∗([0, x]), x > 0

whereWn∗(dx) is the n-fold convolution measure of a unique positive measureW(dx) such that

1f (x)=

∫ ∞

0e−sxW(ds), x > 0

where f (x) is a Bernstein function, satisfying the conditions

limx↓0

f (x) = 0 and limx→∞

f (x) = ∞.


A distribution is said to have complete monotone derivativeif its distributionfunctionF(x) is Bernstein. Pillai and Sandhya (1990) proved that the class of dis-tributions having complete monotone derivative is a propersubclass of g.i.d. dis-tributions. This implies that all distributions with complete monotone densities aregeometrically infinitely divisible. It is easier to verify the complete monotone cri-terion and using this approach we can establish the geometric infinite divisibility ofmany distributions such as Pareto, gamma and Weibull.

The class of non–degenerate generalized gamma convolutions with densities ofthe form given by

f (x) = c xβ−1M∏

j=1

(1+ cj x)−r j , x > 0

is geometrically infinitely divisible for 0< β ≤ 1. Similarly distributions havingdensities of the form

f (x) = cxβ−1 exp(−cxα); 0 < α ≤ 1

is g.i.d. for 0< β ≤ 1. Also the Bondesson family of distributions with densities ofthe form

f (x) = cxβ−1M∏

j=1

1+

Nj∑

k=1

cjkxα jk

−r j

is g.i.d. for 0≤ β ≤ 1, α jk ≤ 1 provided all parameters are strictly positive (seeBondesson(1992)).

4.3.4. Self–decomposability

Let {xn; n ≥ 1} be a sequence of independent random variables, and let{bn} bea sequence of positive real numbers such that

limn→∞

max1≤k≤n

P{|xk| ≥ bnǫ} = 0 for everyǫ > 0.

Let sn =∑n

k=1 xk for n ≥ 1. Then the class of distributions which are the weaklimits of the distributions of the sumsb−1

n sn − an; n ≥ 1 wherean andbn > 0 aresuitably chosen constants, is said to constitute classL. Such distributions are calledself–decomposable.


A distributionF with characteristic functionϕ(t) is called self–decomposable,if and only if, for everyα ∈ (0, 1), there exists a characteristic functionϕα(t) suchthatϕ(t) = ϕ(αt)ϕα(t) for t ∈ R.

Clearly, apart fromx ≡ 0, no lattice random variable can be self–decomposable.All non-degenerate self–decomposable distributions are absolutely continuous.

A discrete analogue of self–decomposability was introduced by Steutel andVan Harn (1979). A distribution onN0 ≡ {0, 1, 2, . . .} with probability gener-ating function (p.g.f.) P(z) is called discrete self–decomposable if and only ifP(z) = P(1− α + αz)Pα(z); |z| ≤ 1,α ∈ (0, 1) wherePα(z) is a p.g.f.

If we defineG(z) = P(1 − z), then G(z) is called the alternate probabilitygenerating function (a.p.g.f.). Then it follows that a distribution is discrete self–decomposable if and only ifG(z) = G(αz)Gα(z); |z| ≤ 1, α ∈ (0, 1) whereGα(z) issome a.p.g.f.

4.3.5. Stable distributions

A distribution functionF with characteristic functionϕ(t) is stable if for everypair of positive real numbersb1 andb2, there exist finite constantsa andb > 0 suchthatϕ(b1t)ϕ(b2t) = ϕ(bt)eiat wherei =

√−1.

Clearly, stable distributions are in classL with the additional condition that therandom variablesxn; n ≥ 1 in Subsection 4.3.4 are identically distributed also.F isstable if and only if its characteristic function can be expressed as

lnϕ(t) = iαt − c|t|β[1 + iγω(t, β)sgn t]

whereα, β, γ are constants withc ≥ 0, 0< β ≤ 2, |γ| ≤ 1 and

ω(t, β) =

tanπβ

2 ; β , 12π

ln |t|; β = 1.

The valuec = 0 corresponds to the degenerate distribution, andβ = 2 to the normaldistribution. The caseγ = 0, β = 1 corresponds to the Cauchy law (see Laha andRohatgi (1979)).

4.3.6. Geometrically strictly stable distributions

A random variabley is said to be geometrically strictly stable (g.s.s.) if for anyp ∈ (0, 1) there exists a constantc = c(p) > 0 and a sequence of independent and


identically distributed random variablesy1, y2, . . . such that

yd= c(p)

N(p)∑

j=1

y j

whereP{N(p) = k} = p(1− p)k−1; k = 1, 2, . . . andy,N(p) andy j; j = 1, 2, . . . areindependent.

If ϕ(t) is the characteristic function ofy, then it implies that

ϕ(t) =pϕ(ct)

1− (1− p)ϕ(ct); p ∈ (0, 1).

Among the geometrically strictly stable distributions, the Laplace distribution andexponential distribution possess all moments. A geometrically strictly stable ran-dom variable is clearly geometrically infinitely divisible.

A non–degenerate random variabley is geometrically strictly stable if and onlyif its characteristic function is of the form

ϕ(t) = 1/ [

1+ λ|t|α exp(

−iπ

2θα sgn t

)]

where 0< α ≤ 2, λ > 0, |θ| ≤ min(1, 2/α − 1). Whenα = 2, it corresponds to theLaplace distribution. Thus it is apparent that when ordinary summation of randomvariables is replaced by geometric summation, the Laplace distribution plays therole of the normal distribution, and exponential distribution replaces the degeneratedistribution (see Klebanovet al.(1984)).

4.3.7. Mittag-Leffler distribution

The Mittag-Leffler distribution was introduced by Pillai (1990a) and has cumu-lative distribution function given by

Fα(x) =∞∑

k=1

(−1)k−1xkα

Γ(1+ kα); 0 < α ≤ 1; x > 0.

Its Laplace transform is given byφ(t) =1

1+ tα; 0 < α ≤ 1; t ≥ 0; and the distribu-

tion may be denoted by ML(α). Hereα is called the exponent. It can be regarded asa generalization of the exponential distribution in the sense thatα = 1, correspondsto the exponential distribution. The Mittag-Leffler distribution is geometrically in-finitely divisible and belongs to classL. It is normally attracted to the stable lawwith exponentα.


If u is exponential with unit mean andy is positive stable with exponentα, thenx = u1/αy is distributed as Mittag-Leffler (α). If u is Mittag-Leffler (α) and v is

exponential andu andv are independent, thenx =uv

is distributed as Pareto type III

with survival functionF̄x(x) = P(x > x) =1

1+ xα; 0 < α ≤ 1.

For the Mittag-Leffler distribution,E(xδ) exists for 0≤ δ < α and is given by

E(xδ) =Γ(1− δ/α)Γ(1+ δ/α)

Γ(1− δ).

A two parameter Mittag-Leffler distribution can also be defined with the cor-

responding Laplace transformφ(t) =λα

λα + tα; 0 < α ≤ 1. It may be denoted by

ML(α, λ).

Jayakumar and Pillai (1993) considered a more general classcalled semi–Mittag-Leffler distribution which included the Mittag-Leffler distribution as a special case.A random variablex with positive support is said to have a semi–Mittag-Leffler dis-tribution if its Laplace transform is given by

φ(t) =1

1+ η(t)

whereη(t) satisfies the functional equationη(t) = aη(bt) where 0< b < 1 andα isthe unique solution ofabα = 1. It may be denoted by SML(α). Then it follows that

η(bt) = bαh(t) whereh(t) is a periodic function int with period− ln b2πα

. Whenh(t)

is a constant, the distribution reduces to the Mittag-Leffler distribution. The semi–Mittag-Leffler distribution is also geometrically infinitely divisibleand belongs toclassL.

4.3.8. α–Laplace distribution

Theα–Laplace distribution has characteristic function given by ϕ(t) =1

1+ |t|α;

0 < α ≤ 2, −∞ < t < ∞. This is also called Linnik’s distribution. Pillai (1985)refers to it as theα–Laplace distribution sinceα = 2 corresponds to the Laplacedistribution. It is unimodal, geometrically strictly stable and belongs to classL. It


is normally attracted to the symmetric stable law with exponentα. Also

E(|x|δ) =2δ Γ

(

1+ δα

)

Γ

(

1− δα

)

Γ((1+ δ)/2)√π Γ

(

1− δ

2

)

where 0< δ < α; 0 < α ≤ 2.

If u andv are independent random variables whereu is exponential with unitmean andv is symmetric stable with exponentα, thenx = u1/αv is distributed asα–Laplace. Using this result, Devroye (1990) develops an algorithm for generatingrandom variables havingα–Laplace distribution.

Pillai (1985) introduced a larger class of distributions called semi–α–Laplacedistribution, with characteristic function given by

ϕ(t) =1

1+ η(t)

whereη(t) satisfies the functional equationη(t) = aη(bt) for 0 < b < 1 anda is theunique solution ofabα = 1, 0 < α ≤ 2. Hereb is called the order andα is calledthe exponent of the distribution. Ifb1 andb2 are the orders of the distribution such

thatln b1

ln b2is irrational, thenη(t) = c|t|α, wherec is some constant. Pillai (1985)

established that, for a semi–α–Laplace distribution with exponentα, E|x|δ exists for0 ≤ δ < α. It can be shown that

ϕ(t) =1

1+ |t|α[1 − Acos (k ln |t|)]

wherek =2π

ln b, 0 < b < 1 is the characteristic function of a semi–α–Laplace

distribution for suitable choice ofA andα < 1.

The semi–α–Laplace distribution is also geometrically infinitely divisible andbelongs to classL. It is useful in modelling household income data. Mohanetal.(1993) refer to it as a geometrically right semi–stable law.

4.3.9. Semi–Pareto distribution

The semi–Pareto distribution was introduced by Pillai (1991). A random vari-able x with positive support has semi–Pareto distributionS P(α, p) if its survivalfunction is given byF̄x(x) = P(x > x0) = 1

1+ψ(x0) whereψ(x0) satisfies the functionalequationpψ(x) = ψ(p1/αx); 0 < p < 1, α > 0.


The above definition is analogous to that of the semi–stable law defined byLevy (see Pillai (1971)). It can be shown thatψ(x) = xαh(x) whereh(x) is periodic

in ln x with period−2παln p

. For example ifh(x) = exp[β cos(α ln x)], then it satisfies

the above functional equation withp = exp(−2π) andψ(x) monotone increasingwith 0 < β < 1. The semi–Pareto distribution can be viewed as a more generalclass which includes the Pareto type III distribution whenψ(x) = cxα, wherec is aconstant.

Exercises 4.3.

4.3.1. Examine whether the following distributions are infinitelydivisible.

(i) normal (ii) exponential (iii) Laplace (iv) Cauchy(v) binomial (vi) Poisson (vii) Geometric (viii) negative binomial

4.3.2. Show that exponential distribution is geometric infinite divisible and self-decomposable.

4.3.3. Examine whether Cauchy distribution is self-decomposable.

4.3.4. Show that (i) Mittag-Leffler distribution is g.i.d. and belongs to class L.(ii) α-Laplace distribution is g.i.d. and self-decomposable.

4.3.5. Give a distribution which is infinitely divisible but not g.i.d.

4.3.6. Show that the AR(1) structurexn = axn−1 + ǫn; a ∈ (0, 1) is stationaryMarkovian if and only if{xn} is self-decomposable.

4.3.7. Obtain the stationary distribution of{ǫn} in the AR(1) structurexn =

axn−1 + ǫn; a ∈ (0, 1) when{xn} follows exponential distribution. Generalize it tothe case of Mittag-Leffler random variables.

4.3.8. Obtain the structure of the innovation distribution if{xn} followsα-Laplacedistribution wherexn = axn−1 + ǫn. Deduce the case whenα = 2.

4.3.9. Show that if{xn} follows Cauchy distribution then{ǫn} also follows aCauchy distribution in the AR(1) equationxn = axn−1 + ǫn.

4.4. STATIONARY TIME SERIES 137

4.3.10. show that geometric and negative binomial distributions are discreteself-decomposable.

4.3.11. Consider the symmetric stable distribution with characteristic functionϕ(t) = e−|t|

α

. Is it self-decomposable ?

4.4. Stationary Time Series

A time series,{xt}, is a family of real–valued random variables indexed byt ∈ �,where� denotes the set of integers. More specifically, it is referred to as a discreteparameter time series. The time series{xt} is said to be stationary if, for anyt1, t2, . . .,tn ∈ �, anyk ∈ �, andn = 1, 2, . . . ,

Fxt1 ,xt2 ,...,xtn(x1, x2, . . . , xn) = Fxt1+k,xt2+k,...,xtn+k(x1, x2, . . . , xn)

whereF denotes the distribution function of the set of random variables whichappear as suffices. This is called stationarity in the strict sense.

Less stringently, we say a process{xn} is weakly stationary if the mean andvariance ofxt remain constant over time and the covariance between any twovaluesxt andxs depends only on the time difference and not on their individual time points.

{xt} is called a Gaussian process if, for alltn; n ≥ 1 the set of random variables{xt1, xt2, . . . , xtn} has a multivariate normal distribution.

Since a multivariate normal distribution is completely specified by its mean vec-tor and covariance matrix, it follows that for a Gaussian process weak stationarityimplies complete stationarity. But for non–Gaussian processes, this may not hold.

4.4.1. Autoregressive models

The era of linear time series models began with autoregressive models first in-troduced by Yule in 1927. The standard form of an autoregressive model of orderp, denoted by AR(p), is given by

xt =

p∑

j=1

a j xt− j + ǫt; t = 0,±1,±2, . . .

where{ǫt} are independent and identically distributed random variables called inno-vations anda j, p are fixed parameters, withap , 0.


Another kind of model of great practical importance in the representation ofobserved time series is the moving average model. The standard form of a moving

average model of orderq, denoted by MA(q), is given byxt =

q∑

j=1b jǫt− j + ǫt; t ∈ �

whereb j , q are fixed parameters, withbq , 0.

To achieve greater flexibility in the fitting of actually observed time series, itis more advantageous to include both autoregressive and moving average terms inthe model. Such models called autoregressive–moving average models, denoted byARMA(p,q), have the form

xt =

p∑

j=1

a j xt− j +

q∑

k=1

bkǫt−k + ǫt; t ∈ �

where{a j}pj=1 and{bk}qk=1 are real constants called parameters of the model. It can beseen that an AR(p) model is the same as an ARMA(p,o) model and aMA(q) modelis the same as an ARMA(o,q) model.

With the introduction of various non–Gaussian and non–linear models, the stan-dard form of autoregression was widened in several respects.

A more general definition of autoregression of orderp is given in terms of thelinear conditional expectation requirement that

E(xt |xt−1, xt−2, . . .) =p∑

j=1

a j xt− j

This definition could apply to models which are not of the linear form (see Lawrance(1991)).

4.4.2. A general solution

We consider a first order autoregressive model with innovation given by thestructural relationship

xn = ǫn +

0 with probabilityp

xn−1 with probability 1− p(4.4.1)

wherep ∈ (0, 1) and{ǫn} is a sequence of independent and identically distributed(i.i.d.) random variables selected in such a way that{xn} is stationary Markovianwith a given marginal distribution functionF.


Let φx(t) = E[e−tx] be the Laplace–Stieltjes transform ofx. Then (4.4.1) gives

φxn(t) = φǫn(t)[p+ (1− p)φxn−1(t)]

If we assume stationarity, this simplifies to

φǫ(t) =φx(t)

p+ (1− p)φx(t)(4.4.2)

or equivalently

φx(t) =pφǫ(t)

1− (1− p)φǫ(t). (4.4.3)

When{xn} is marginally distributed as exponential, it is easy to see that (4.4.1) givesthe TEAR(1) model.

We note thatφǫ(t) in (4.4.2) does not represent a Laplace transform always. Inorder that the process given by (4.4.1) is properly defined, there should exist aninnovation distribution such thatφǫ(t) is a Laplace transform for allp ∈ (0, 1). Toestablish the main results we need the following lemmas.

Lemma 4.4.1. (Pillai (1990b)) LetF be a distribution with positive support andφ(t) beits Laplace transform. ThenF is geometrically infinitely divisible if and only if

φ(t) =1

1+ ψ(t)

whereψ(t) is Bernstein withψ(0) = 0.

Now we consider the following definition from Pillai (1990b).

Definition 4.4.1. For any non–vanishing Laplace transformφ(t), the functionψ(t) =1φ(t)− 1 is called the third characteristic.

Lemma 4.4.2. Let ψ(t) be the third characteristic ofφ(t). Thenpψ(t) is a third character-istic for all p ∈ (0, 1) if and only ifψ(t) has complete monotone derivative andψ(0) = 0.

Thus we have the following theorem.

Theorem 4.4.1. φǫ(t) in (4.4.2) represents a Laplace transform for all p∈ (0, 1)if and only ifφx(t) is the Laplace transform of a geometrically infinitely divisibledistribution.


This leads to the following theorem which brings out the roleof geometrically infinitelydivisible distributions in defining the new first order autoregressive model given by (4.4.1).

Theorem 4.4.2. The innovation sequence{ǫn} defining the first order autoregres-sive model given by

xn = ǫn +

0 with probability p

xn−1 with probability1− p

where p∈ (0, 1), exists if and only if the stationary marginal distributionof xn isgeometrically infinitely divisible. Then the innovation distribution is also geometri-cally infinitely divisible.

Proof. Suppose that an innovation sequence{ǫn} such that the model (4.4.1) is properlydefined exists. This implies thatφǫ(t) in (4.4.2) is a Laplace transform for allp ∈ (0, 1).Then from (4.4.3)

φx(t) = pφǫ (t)[1 − (1− p)φǫ(t)]−1

=

∞∑

n=1

p(1− p)n−1[φǫ (t)]n

showing that the stationary marginal distribution ofxn is geometrically infinitely divisible.Conversely, ifxn has a stationary marginal distribution which is geometrically infinitely

divisible, thenφx(t) =1

1+ ψ(t)whereψ(t) has complete monotone derivative andψ(0) = 0.

Then from (4.4.2) we getφǫ(t) =1

1+ pψ(t), which establishes the existence of an innovation

distribution, which is geometrically infinitely divisible.

4.4.3. Extension to a k-th order autoregressive modelIn this section we consider an extension of the model given by(4.4.1) to thek-th order.

The structure of this model is given by

xn = ǫn +

0 with probability p0

xn−1 with probability p1...

xn−k with probability pk

(4.4.4)


wherepi ∈ (0, 1) for i = 0, 1, . . . , k andp0 + p1 + · · · + pk = 1. Taking Laplace transformson both sides of (4.4.4) we get

φxn(t) = φǫn(t)

p0 +

k∑

i=1

piφxn−i (t)

Assuming stationarity, it simplifies to

φx(t) = φǫ(t)

p0 +

k∑

i=1

piφx(t)

= φǫ(t)[p0 + (1− p0)φx(t)].

This yields

φǫ(t) =φx(t)

p0 + (1− p0)φx(t)(4.4.5)

which is analogous to the expression (4.4.2).

It may be noted thatk = 1 corresponds to the first order model withp = p0. From(4.4.6) it follows that the results obtained in Section 4.4.2 hold good for thek-th ordermodel given by (4.4.5). This establishes the importance of geometrically infinitely divisibledistributions in autoregressive modelling.

4.4.4. Mittag-Leffler autoregressive structure

The Mittag-Leffler distribution was introduced by Pillai (1990a) and has Laplace trans-

form φ(t) =1

1+ tα, 0 < α ≤ 1. Whenα = 1, this corresponds to the exponential distribution

with unit mean. Jayakumar and Pillai (1993) considered the semi–Mittag-Leffler distribu-

tion with exponentα. Its Laplace transform is of the form1

1+ η(t)whereη(t) satisfies the

functional equationη(t) = aη(bt), 0 < b < 1 (4.4.6)

and a is the unique solution ofabα = 1 where 0< α ≤ 1. Then by Lemma 4.2.1 ofJayakumar and Pillai (1993), the solution of the functionalequation (4.4.6) isη(t) = tαh(t)

whereh(t) is periodic in lnt with period−2παln b

. Whenh(t) = 1, η(t) = tα and hence the

Mittag-Leffler distribution is a special case of the semi-Mittag-Leffler distribution. It isobvious that the semi–Mittag-Leffler distribution is geometrically infinitely divisible.

Now we bring out the importance of the semi-Mittag-Leffler distribution in the contextof the new autoregressive structure given by (4.4.1). The following theorem establishes this.


Theorem 4.4.3. For a positive valued first order autoregressive process{xn} satis-fying(4.4.1) the stationary marginal distribution of xn andǫn are identical except fora scale change if and only if xn’s are marginally distributed as semi-Mittag-Leffler .

Proof. Suppose that the stationary marginal distributions ofxn andǫn are identical. Thisimpliesφǫ(t) = φx(ct) wherec is a constant. Then from (4.4.2) we get

φx(ct) =φx(t)

p+ (1− p)φx(t)(4.4.7)

Writing φx(t) =1

1+ η(t)in (4.4.7) we get

11+ η(ct)

=1

1+ pη(t)

so that

η(ct) = pη(t)

By choosingc = p1/α, it follows thatxn is distributed as semi–Mittag-Leffler with exponentα.

Conversely, we assume that the stationary marginal distribution of xn is semi–Mittag-Leffler . Then from (4.4.2)

φǫ(t) =1

1+ pη(t)=

1

1+ η(p1/αt).

This establishes thatǫnd= p1/αMn where{Mn} are independently and identically distributed

as semi–Mittag-Leffler .It can be easily seen that the above result is true in the case of thek-th order autoregres-

sive model given by (4.4.4) also.

Exercises 4.4.

4.4.1. Consider a Poisson process{x(t)} wherep[x(t) = n] = e−λt(λt)n

n! ; n = 0, 1, · · ·Find E(x(t)) and Var(x(t)). Is the process stationary?

4.4.2. Consider a Poisson process{x(t)} as above. Letx0 be independent ofx(t) such thatp(x0 = 1) = p(x0 = −1) = 1

2.DefineN(t) = x0(−1)N(t). FindE(N(t)) and Cov(N(t),N(t+s)).

4.4.3. Define an AR(1) process and obtain the stationary solution for the distribution of{ǫn} when{xn} are exponentially distributed.

4.5. A STRUCTURAL RELATIONSHIP 143

4.4.4. Show that an AR(1) model can be expressed as anMA(∞) model.

4.4.5. Construct a new AR(1) model with exponential innovations.

4.4.6. Examine whether the two-parameter Gamma distribution is g.i.d., giving conditionsif any.4.4.7. Show that exponential distribution is a special case of Mittag-Leffler distribution.

4.5. A Structural RelationshipIn this section we obtain the specific structural relationship between the stationary mar-

ginal distributions ofxn andǫn in the new autoregressive model.

Fujita (1993) generalized the results on Mittag-Leffler distributions and obtained anew characterization of geometrically infinitely divisible distributions with positive supportusing Bernstein functions. It was established that a distribution functionG with G(0) = 0 isgeometrically infinitely divisible if and only ifG can be expressed in the form.

G(x) =∞∑

n=1

(−1)n+1λnWn∗([0, x]); x > 0, λ > 0 (4.5.1)

whereWn∗(dx) is then-fold convolution measure of a unique positive measureW(dx) on[0,∞) such that

1f (x)=

∫ ∞

0e−sxW(ds); x > 0 (4.5.2)

for some Bernstein functionf such that limx↓o f (x) = 0 and limx→∞ f (x) = ∞. Then the

Laplace transform ofG(x) isλ

λ + f (t). Using this result we get the following theorem.

Theorem 4.5.1. The k-th order autoregressive equation given by (4.4.4) defines astationary process with a given marginal distribution function Fx(x) for xn if andonly if Fx(x) can be expressed in the form

Fx(x) =∞∑

n=1

(−1)n+1λnWn∗([0, x]); x > 0, λ > 0. (4.5.3)

Then the innovations{ǫn} have a distribution function Fǫ(x) given by

Fǫ(x) =∞∑

n=1

(−1)n+1(λ/p0)nWn∗([0, x]); x > 0, λ > 0, (4.5.4)

where p0 ∈ (0, 1) and Wn∗ is as in(4.5.1).


Proof. We have from Theorem 4.4.1 thatFx(x) is geometrically infinitely divisible. Then(4.5.3) follows directly from Fujita (1993).

Now by substitutingφx(t) =λ

λ + f (t)in (4.3.2) we get

φǫ(t) =λ

λ + p0 f (t)=

(λ/p0)(λ/p0) + f (t)

which leads to (4.5.4). This completes the proof.

The above theorem can be used to construct various autoregressive models under dif-ferent stationary marginal distributions forxn.

For example, the TEAR(1) model of Lawrance and Lewis (1981) can be obtained by

taking f (t) = t. ThenWn∗([0, x]) =xn

n!so thatFx(x) = 1 − e−λx andFǫ(x) = 1 − e−(λ/p)x.

If we takeλ = 1 and f (t) = tα; 0 < α ≤ 1 we can obtain an easily tractable first orderautoregressive Mittag-Leffler process denoted by TMLAR(1). In this caseWn∗([0, x]) =

xnα

Γ(1+ nα). In a similar manner by takingλ = 1 and f (t) satisfying the functional equation

f (t) = a f(bt) wherea = b−α; 0 < b < 1, 0< α ≤ 1, we can obtain an easily tractable firstorder autoregressive semi–Mittag-Leffler process denoted by TSMLAR(1).

4.5.1. The TMLAR(1) processAn easily tractable form of a first order autoregressive Mittag-Leffler process, called

TMLAR(1), is constituted by{xn} having a structure of the form

xn = p1/αMn +


xn−1 with probability 1-p(4.5.5)

where p ∈ (0, 1); 0 < α ≤ 1 and {Mn} is independently and identically distributed as

Mittag-Leffler with exponentα and x0d= M1. The model (4.5.5) can be rewritten in the

form

xn = p1/αMn + Inxn−1 (4.5.6)

where{In} is a Bernoulli sequence such thatP(In = 0) = p andP(In = 1) = 1− p.

If in the structural form (4.5.5), we assume that{Mn} are distributed as semi–Mittag-Leffler with exponentα, then{xn} constitute a tractable semi–Mittag-Leffler autoregressiveprocess of order 1, called TSMLAR(1). Both models are Markovian and stationary. It canbe seen that the TMLAR(1) process is a special case of the TSMLAR(1) process since theMittag-Leffler distribution is a special case of the semi–Mittag-Leffler distribution.

4.5. A STRUCTURAL RELATIONSHIP 145

Now we shall consider the TSMLAR(1) process and establish that it is strictly stationaryand Markovian, providedx0 is distributed as semi–Mittag-Leffler . In order to prove this weuse the method of induction.

Suppose thatxn−1 is distributed as semi–Mittag-Leffler (α). Then by taking Laplacetransforms on both sides of (4.5.5), we get

φxn(t) = φMn(p1/αt)[p+ (1− p)φxn−1(t)]

=1

1+ η(p1/αt)

[

p+ (1− p)1

1+ η(t)

]

=1

1+ pη(t)·

1+ pη(t)1+ η(t)

=1

1+ η(t).

Hencexn is distributed as semi–Mittag-Leffler with exponentα.

If x0 is arbitrary, then also it is easy to establish that{xn} is asymptotically stationary.Thus we have the following theorem.

Theorem 4.5.2. The first order autoregressive equation

xn = p1/αMn + Inxn−1; n = 1, 2, . . . , p ∈ (0, 1)

where {In} are independent Bernoulli random variables such thatP(In = 0) = p = 1 − P(In = 1) defines a positive valued strictly stationary firstorder autoregressive process if and only if{Mn} are independently and identically

distributed as semi–Mittag-Leffler with exponentα and x0d= M1.

Remark If we consider characteristic functions instead of Laplacetransforms, the resultscan be applied to real valued autoregressive processes. Then the role of semi–Mittag-Lefflerdistributions is played by semi–α–Laplace distributions introduced by Pillai (1985).

4.5.2. The NEAR(1) model

In this section we consider a generalized form of the first order autoregressive equation.The new structure is given by

xn = ǫn +


axn−1 with probability 1− p(4.5.7)


where 0≤ p ≤ 1; 0≤ a ≤ 1 and{ǫn} is a sequence of independent and identically distributedrandom variables such that{xn} have a given stationary marginal distribution. Letφx(t) =E[e−tx] be the Laplace–Stieltjes transform ofx. Then (4.5.7) gives

φxn(t) = φǫn(t)[p+ (1− p)φxn−1(at)]

Assuming stationarity, it simplifies to

φǫ(t) =φx(t)

p+ (1− p)φx(at). (4.5.8)

Whenp = 0 and 0< a < 1, the model (4.5.7) is the standard first order autoregressivemodel. Then the model is properly defined if and only if the stationary marginal distributionof xn is self–decomposable. Whena = 1, 0 < p < 1 the model is the same as the model(4.4.1), which is properly defined if and only if the stationary marginal distribution ofxn isgeometrically infinitely divisible. Whena = 0 or p = 1, xn andǫn are identically distributed.

Now we consider the case whena ∈ (0, 1] and p ∈ (0, 1], but not simultaneouslyequal to 1. Lawrance and Lewis (1981) developed an NEAR(1) model with exponential (λ)

marginal distribution forxn. Thenφx(t) =λ

λ + tand substitution in (4.5.8) gives

φǫ(t) =λ + atλ + t

· λ

λ + pat(4.5.9)

which can be rewritten as

φǫ(t) =

(

1− a1− pa

) (λ

λ + t

)

+

[

(1− p)a1− pa

] (

λ

λ + pat

)

.

Henceǫn can be regarded as a convex exponential mixture of the form

ǫn =

En with probability 1−a1−pa

paEn with probability (1−p)a1−pa

(4.5.10)

where {En}; n = 1, 2, . . . are independent and identically distributed as exponential (λ)random variables. Another representation forǫn can be obtained from (4.5.9) by writing

φǫ(t) =[

a+ (1− a)λ

λ + t

][λ

λ + pat

]

. (4.5.11)

Then writing w.p. for ‘with probability’ǫn can be regarded as the sum of two independentrandom variablesun andvn where

un =

0 w. p. a

En w. p. 1− a and vn = paEn(4.5.12)

4.6. NEW MITTAG-LEFFLER AUTOREGRESSIVE MODELS 147

where{En}; n = 1, 2, . . . are exponential (λ). It may be noted that whenp = 0, the model isidentical with the EAR(1) process, of Gaver and Lewis (1980). Thus the new representationof ǫn seems to be more appropriate, when NEAR(1) process is regarded as a generalizationof the EAR(1) process.

Exercises 4.5.

4.5.1. If f (t) = t, find Wn∗([0, x]).

4.5.2. If f (t) = tα, find Fǫ(x).

4.5.3. State any three distributions belonging to the semi-Mittag-Leffler family.

4.5.4. Show that the stationary solution of Equation 4.5.7 is a family consisting of g.i.d.and class L distributions.

4.5.5. Obtain the innovation structure of the NEAR(1) model.

4.5.6. Obtain the innovation structure of the NMLAR(1) model.

4.6. New Mittag-Leffler Autoregressive ModelsNow we construct a new first order autoregressive process with Mittag-Leffler marginal

distribution, called the NMLAR(1) model. The structure of the model is as in (4.5.7) and

the innovations can be derived by substitutingφx(t) =1

1+ tα; 0 < α ≤ 1 in (4.5.8). This

gives

φǫ(t) =1+ aαtα

1+ tα· 1

1+ aαptα.

Hence the innovationsǫn can be given in the form

ǫn =

Mn with probability 1−aα

1−paα

paαMn with probability (1−p)aα

1−paα(4.6.1)

where{Mn} are Mittag-Leffler (α) random variables.

An alternate representation ofǫn is ǫn = un+vn whereun andvn are independent randomvariables such that

un =

0 w.p. aα

Mn w.p. 1− aα andvn = ap1/αMn(4.6.2)


where{Mn}; n = 1, 2, . . . are independent Mittag-Leffler (α) random variables.

It can be shown that the process is strictly stationary and Markovian. This gives us thefollowing theorem.

Theorem 4.6.1. The first order autoregressive equation given by(4.5.7) definesa strictly stationaryAR(1) process with a Mittag-Leffler (α) marginal distributionfor xn if and only if the innovations are of the formǫn = un + vn where un andvn areas in(4.6.2) with x0 distributed as Mittag-Leffler (α).

Proof. We prove this by induction. We assume thatxn−1 is Mittag-Leffler (α). Then bytaking Laplace transforms, we get

φxn(t) = φMn(ap1/αt) · [aα + (1− aα)φMn(t)]

× [p+ (1− p)φxn−1(at)]

=1

1+ aαptα·[

aα + (1− aα)1

1+ tα

]

×[

p+ (1− p)1

1+ aαtα

]

=1

1+ aαptα·

1+ aαtα

1+ tα·

1+ paαtα

1+ aαtα

=1

1+ tα.

This shows thatxn is distributed as Mittag-Leffler (α), and this establishes the sufficiencypart.

The necessary part is obvious from the derivation of the innovation sequence. Thiscompletes the proof.

The joint distribution of (xn, xn−1) is of interest in describing the process and matching itwith data. Therefore, we shall obtain the joint distribution with the use of Laplace-Stieltjestransforms. The bivariate Laplace transform is given by

φxn,xn−1(s, t) = E{exp(−sxn − txn−1)}= φǫ(s){pφx(t) + (1− p)φx(as+ t)}

=1+ aαsα

1+ sα· 1

1+ paαsα·{

p1+ tα

+1− p

1+ (as+ t)α

}

.

It is possible to obtain the joint distribution by invertingthis expression.


4.6.1. The NSMLAR(1) processNow we extend the NMLAR(1) process to a wider class to construct a new semi–

Mittag-Leffler first order autoregressive process. The process has the structure

xn = ǫn +


axn−1 with probability 1− p

where {ǫn} are independently and identically distributed as the sum oftwo independentrandom variablesun andvn where

un =

0 w.p. aα

Mn w.p. 1− aα andvn = ap1/αMn(4.6.3)

where {Mn}; n = 1, 2, . . . are independently and identically distributed as semi–Mittag-Leffler (α).

This process is also clearly strictly stationary and Markovian providedx0 is semi–Mittag-Leffler (α). This follows by induction. In terms of Laplace transformswe have

φxn(t) = [p+ (1− p)φxn−1(at)][aα + (1− aα)φMn(t)]

· [φMn(ap1/αt)]

=

[

p+ (1− p) · 11+ η(at)

] [

aα + (1− aα) · 11+ η(t)

]

·1

1+ η(ap1/αt)

=

[

p+ (1− p)1

1+ aαη(t)

]

·[

1+ aαη(t)1+ η(t)

]

·1

1+ aαpη(t)

=1

1+ η(t).

Thus we have established the following theorem.

Theorem 4.6.2. The first order autoregressive equation

xn = aInxn−1 + ǫn; n = 1, 2, . . .

where{In} are independent Bernoulli sequences such that P(In = 0) = p and P(In =

1) = 1 − p; p ∈ (0, 1), a ∈ (0, 1) is a strictly stationary AR(1) process with semi–Mittag-Leffler (α) marginal distribution if and only if{ǫn} are independently and


identically distributed as the sum of two independent random variables un andvn asin (3.8.1) and x0 is distributed as semi–Mittag-Leffler (α).

Whenη(t) = tα, the NSMLAR(1) model becomes the NMLAR(1) model.

4.6.2. Semi–Mittag-Leffler tailed processes

In an attempt to develop autoregressive models for time series with exact zeroes Little-john (1993) formulated an autoregressive process with exponential tailed marginal distribu-tion, after the new exponential autoregressive process (NEAR(1)) of Lawrance and Lewis(1981). However, the primary aim of Littlejohn was to extendthe time reversibility theoremof Chernicket al.(1988) and hence the model was not studied in detail. Hence weintendto make a detailed study on this process. Here the tail of a non–negative random variablerefers to the positive part of the sample space, excluding only the point zero.

Definition 4.6.1. A random variableE is said to have the exponential tailed dis-tribution denoted byET(λ, θ) if P(E = 0) = θ andP(E > x) = (1 − θ)e−λx; x > 0whereλ > 0 and 0≤ θ < 1. Then the Laplace-Stieltjes transform ofE is given by

φE(t) = θ + (1− θ) λ

λ + t

=λ + θtλ + t

4.6.3. The exponential tailed autoregressive process [ETAR(1)]

It is evident that the exponential tailed distribution is not self–decomposable and soit cannot be marginal to the autoregressive structure of Gaver and Lewis (1980). But anautoregressive process satisfying the NEAR(1) structure given by (4.5.7) can be constructedas follows.

We have from (4.5.8), by substitutingφx(t) =λ + θtλ + t

, the Laplace transform of the

innovationǫn in the stationary case as

φǫ(t) =[λ + θtλ + t

] [

λ + atλ + a[p+ (1− p)θ]t

]

=

[λ + atλ + t

] [λ + θtλ + bt

]


whereb = a[p+ (1− p)θ]

φǫ(t) =[

a+ (1− a)λ

λ + t

] [

θ

b+

(

1−θ

b

) (λ/b)(λ/b) + t

]

so that the innovations{ǫn} can be represented as the sum of two independent exponentialtailed random variablesun andvn where

und= ET(λ, a) and vn

d= ET

(λ′, θ′

)(4.6.4)

whereλ′ = λ/b andθ′ = θ/b, providedθ ≤ b. Sincep ≤ 1, we require thatθ ≤ a. Thusthe ETAR(1) process can be defined as a sequence{xn} satisfying (4.5.3) where{ǫn} is asequence of independent and identically distributed random variables such thatǫn = un + vnwhereun andvn are as in (4.6.4).

It can be easily shown that the process is strictly stationary and Markovian providedx0

is distributed asET(λ, θ). This follows by mathematical induction since

φxn(t) = φǫn(t) · [p+ (1− p)φxn−1(at)]

=λ + atλ + t

·λ + θtλ + bt

·[

p+ (1− p)(λ + θatλ + at

)]

=λ + atλ + t

·λ + θtλ + bt

·λ + btλ + at

=λ + θtλ + t

.

Whenθ = 0, theET(λ, θ) distribution reduces to the exponential (λ) distribution and theETAR(1) model then becomes the NEAR(1) model.

4.6.4. The Mittag-Leffler tailed autoregressive process [MLTAR(1)]

The Mittag-Leffler tailed distribution has Laplace transform given by

φx(t) = θ + (1− θ) · 11+ tα

=1+ θtα

1+ tα; 0 < α ≤ 1

and the distribution shall be denoted by MLT (α, θ). Similarly for a two-parameter Mittag-Leffler random variable ML(α, λ) the Laplace transform of the tailed Mittag-Leffler distri-bution is given byφx(t) = θ + (1− θ) λα

λα+tα =λα+θtαλα+tα . This shall be denoted by MLT(α, λ, θ).


The MLTAR(1) process has the general structure given by the equation (4.5.7). The inno-vation structure can be derived as follows.

φǫ(t) =1+ θtα

1+ tα· 1+ aαtα

1+ aα[p+ (1− p)θ]tα

=1+ aαtα

1+ tα·

1+ θtα

1+ ctα

wherec = aα[p+ (1− p)θ]. Therefore

φǫ(t) =

[

aα + (1− aα)1

1+ tα

]

1c +

θctα

1c + tα

.

Hence the innovation{ǫn} can be viewed as the sum of two independently distributed randomvariablesun andvn where

und= MLT(α, aα)

and

vnd= MLT

(α, λ′, θ′

)

whereλ′ = 1/c1/α andθ′ = θ/c providedθ ≤ c. This holds whenθ ≤ aα.

The model can be extended to the class of semi–Mittag-Leffler distributions. Here weconsider a semi–Mittag-Leffler distribution with Laplace transform

φx(t) =λα

λα + η(t)

whereη(t) satisfies the functional equation

η(mt) = mαη(t); 0 < m< 1; 0< α ≤ 1.

This is denoted by SML(α, λ). Then the semi–Mittag-Leffler tailed distribution denoted bySMLT(α, λ, θ) has Laplace transform

φx(t) =λα + θη(t)λα + η(t)

.

The first order semi-Mittag-Leffler tailed autoregressive (SMLTAR(1)) process has innova-tions whose Laplace transform is given by

φǫ(t) =

[

λα + θη(t)λα + η(t)

] [

λα + aαη(t)λα + cη(t)

]


wherec = aα[p+ (1− p)θ]. Therefore

φǫ(t) =

[

λα + aαη(t)λα + η(t)

] [

λα + θη(t)λα + cη(t)

]

=

[

aα + (1− aα)λα

λα + η(t)

] [

θ

c+

(

1− θc

)λα/c

λα/c+ η(t)

]

.

Therefore, the innovations{ǫn} can be represented as the sum of two independent semi–Mittag-Leffler tailed random variablesun andvn where

und= S MLT(α, λ, aα) and vn

d= S MLT

(α, λ′, θ′

)(4.6.5)

whereλ′ = λ/c1/α, θ′ = θ/c. Then we have the following theorem which gives the stationarysolution of the SMLTAR(1) model.

Theorem 4.6.3. For 0 < p < 1, 0 < a < 1 the stationary Markov process{xn}defined by(4.5.7) has a semi–Mittag-Leffler tailed SMLT(α, λ, θ) marginal distri-bution if and only if the innovation sequence{ǫn} are independent and identicallydistributed as the sum of two independent semi–Mittag-Leffler Tailed random vari-

ables as in(4.6.5), provided x0d= S MLT(α, λ, θ).

The stationarity of the process can be easily established, as given below.

φxn(t) = φǫn(t)[p+ (1− p)φxn−1(at)]

=

[


] [


]

×[

p+ (1− p)λα + θη(at)λα + η(at)

]

=

[


] [


] [

λα + cη(t)λα + η(at)

]

=λα + θη(t)λα + η(t)

sinceη(at) = aαη(t).

Hencexn is distributed as SMLT(α, λ, θ). The necessity part follows easily from the deriva-tion of the structure of the innovation sequence. Now we consider the following theorem.

Theorem 4.6.4. In a positive valued stationary Markov process{xn} satisfying thefirst order autoregressive equation xn = axn−1 + ǫn, 0 < a < 1 the innovations{ǫn}are independently and identically distributed as a tailed distribution of the sametype as that of{xn} if and only if{xn} are distributed as semi–Mittag-Leffler .


Proof We have, assuming stationarity,

φx(t) = φx(at)φǫ (t).

Suppose

φǫ(t) = θ + (1− θ)φx(t) where 0≤ θ < 1.

Then

φx(t) = φx(at)[θ + (1− θ)φx(t)].

Writing

φx(t) =1

1+ η(t), we get

11+ η(t)

=1

1+ η(at)

[

θ + (1− θ)1

1+ η(t)

]

=

[

11+ η(at)

] [

1+ θη(t)1+ η(t)

]

.

This impliesη(at) = θη(t). By taking θ = aα, this means that the distribution ofxn is

semi–Mittag-Leffler .

Conversely, if{xn} are semi–Mittag-Leffler , we get

φǫ(t) =φx(t)φx(at)

=1+ η(at)1+ η(t)

=1+ aαη(t)1+ η(t)

= aα + (1− aα)1

1+ η(t).

Hence{ǫn} is distributed as SMLT(α, aα).

The SMLTAR(1) process can be regarded as generalizations ofthe EAR(1), NEAR(1),MLAR(1), NMLAR(1), TEAR(1), ETAR(1) and MLTAR(1) processes. These processesare useful to model non-negative time series data which exhibit zeros, as in the case ofstream flow data of rivers that are dry during part of the year.They are useful for modellinglife times of devices which have some probability for damageimmediately when it is put touse.

Now we extend our results to the case of real valued processes, in which case the timeseries data can be negative as well as positive, including zeros.


4.6.5. Semi–α–Laplace tailed first order autoregressive process[α–SLTAR(1)]

In this section we consider continuous random variables defined over (−∞,∞) and con-struct a new random variable called a tailed random variablewith an atom of massθ atzero and the rest of the probability distributed over the sample space of the original randomvariable excluding the point zero. If the characteristic function of the original random vari-able isϕ(t) then the characteristic function of the new tailed random variable is given byθ + (1− θ)ϕ(t).

Here we construct a time series model with stationary marginal distribution as a semi–

α–Laplace tailed distribution with characteristic function θ+ (1− θ) 11+ η(t)

whereβαη(t) =

η(βt); 0 < β < 1, 0 < α ≤ 2. This distribution shall be denoted by SALT(α, θ) and thecorresponding first order semi–α–Laplace tailed autoregressive process byα-SLTAR(1).

The structure of the model is given by (4.5.7). The characteristic function of the inno-vation sequence{ǫn}, under stationarity, is given by

ϕǫ(t) =ϕx(t)

p+ (1− p)ϕx(at)

=

[

1+ θη(t)1+ η(t)

] [

1+ η(at)1+ cη(t)

]

wherec = aα[p+ (1− p)θ]. Therfore

ϕǫ(t) =

(

1+ aαη(t)1+ η(t)

)

1c +

θcη(t)

1c + η(t)

=

[

aα + (1− aα)1

1+ η(t)

]

θ

c+

(

1−θ

c

)

·1c

1c + η(t)

=

[

aα + (1− aα)1

1+ η(t)

] [

θ

c+

(

1−θ

c

) 1

1+ η(c1/αt)

]

.

Therefore,ǫn can be regarded as the sum of two independent semi–α–Laplace tailedrandom variablesun andvn such that

und= S ALT(α, aα) and vn

d= c1/αS ALT(α, θ/c). (4.6.6)

The model is strictly stationary and Markovian. We have the following theorem.

Theorem 4.6.5. For 0 < p < 1, 0 < a < 1 the stationary Markov process{xn}defined by(4.5.7) has a semi–α–Laplace tailed marginal distribution if and only if


the innovation sequence{ǫn} are independent and identically distributed as the sumof two independent semi–α–Laplace tailed random variables having structure given

by (4.6.6), provided x0d= S ALT(α, θ).

4.6.6. A first order α–Laplace tailed autoregressive [α-LTAR(1)]process

It can be seen that anα–Laplace tailed first order autoregressive process having structure(4.5.7) can be constructed for anya such that|a| ≤ 1. In this case, the characteristic functionof xn is

ϕxn(t) = θ + (1− θ)1

1+ |t|α=

1+ θ|t|α

1+ |t|α; 0 < α ≤ 2.

Then (4.5.8) can be given in terms of characteristic function as

ϕǫ(t) =

[

1+ θ|t|α

1+ |t|α

] [

1+ |at|α

1+ c|t|α

]

wherec = |a|α[p+ (1− p)θ]; 0 < α ≤ 2. Thereforeϕǫ(t) can be written as

ϕǫ(t) =

{

|a|α + (1− |a|α) ·1

1+ |t|α

} {

θ

c+

(

1−θ

c

) 11+ c|t|α

}

.

This shows that the innovations can be written as the sum of two independentα–Laplacetailed random variables of which one has an atom of mass|a|α at zero and the other has anatom of massθc at zero. The corresponding model may be called anα–Laplace tailed firstorder autoregressive process denoted byα–LTAR(1). The model is defined whenθ ≤ |a|α;0 < α ≤ 2, |a| < 1. Whenθ = 0, we get

ϕǫ(t) =

[

|a|α + (1− |a|α)1

1+ |t|α

] [

11+ c|t|α

]

.

This gives a newα–Laplace autoregressive process with innovations of the form ǫn = un+vnwhereun andvn are indpendently distributed as

un =

0 w.p. |a|α

Ln w.p. 1− |a|α andvn = c1/αLn

where{Ln} areα–Laplace variables. This model may be called a newα–Laplace autore-gressive process of order one denoted byα–NLAR(1).


Whenp = 0, this model gives anα–Laplace autoregressive process denoted byα–LAR(1)with the usual linear additive structure. Then the model is given by

xn = axn−1 + ǫn

= axn−1 +

0 with probability |a|α

Ln with probability 1− |a|α

provided|a| ≤ 1; 0< α ≤ 2. Now we have the following theorem:

Theorem 4.6.6. The first order difference equation

xn = axn−1 +

0 with probability |a|α

Ln with probability 1− |a|α(4.6.7)

where|a| < 1, 0 < α ≤ 2 defines a strictly stationary Markovian first order autore-gressive process withα–Laplace marginal distribution, if and only if the innovations{Ln} are independent and identically distributed asα–Laplace and are independent

of x0d= α–Laplace.

Proof. Assuming that the characteristic function ofxn is denoted byϕxn(t), we have from(4.6.7)

ϕxn(t) = ϕxn−1(at){|a|α + (1− |a|α)ϕLn(t)}

Assuming stationarity

ϕx(t) = ϕx(at){|a|α + (1− |a|α)ϕL(t)}

Therefore,

ϕL(t) =ϕx(t) − |a|αϕx(at)(1− |a|α)ϕx(at)

.

Substitutingϕx(t) =1

1+ |t|αwe get

ϕL(t) =1

1+ |t|α.

This establishes the necessary part. The converse is provedby mathematical induction.

Assume thatxn−1d= α–Laplace. Then


ϕxn(t) =1

1+ |at|α

{

|a|α + (1− |a|α)1

1+ |t|α

}

=1

1+ |at|α1+ |a|α|t|α

1+ |t|α

=1

1+ |t|α.

Hencexn is distributed asα–Laplace.

If x0 is arbitrary, then also{xn} is stationary and asymptotically distributed asα–Laplace.This follows since

ϕxn(t) = ϕxn−1(at)

[

1+ |a|α |t|α

1+ |t|α

]

=1+ |a|αn

1+ |t|αϕx0(a

nt)

Hence

ϕxn(t)→1

1+ |t|αas n→ ∞.

Remark The model given by (4.6.7) extends the EAR(1) model of Gaver and Lewis (1980)and the MLAR(1) model of Jayakumar and Pillai (1993). Whenα = 2, the variables aredistributed as double exponential (Laplace). This model can be used for modelling datarelating to stock returns and commodity prices. Rachev and Sen Gupta (1991) describe theuse of geometric stable distributions, which is exactly thesame asα–Laplace distributions,in modelling stock returns. In the class of geometric stabledistributions, Laplace distribu-tion plays the role of normal distribution in the class of stable laws. Anderson and Arnold(1993) also describe the use of the Linnik distribution, which is the same as theα–Laplacedistribution, in modelling financial time series.

Exercises 4.6.

4.6.1. Derive the Laplace transform of the exponential tailed distribution.

4.6.2. Derive the innovation structure of the Mittag-Leffler tailed autoregressive process.

4.6.3. Examine whether the Mittag-Leffler tailed distribution is self-decomposable.


4.6.4. Give a real life example where exponential tailed distribution can be used formodelling.

4.6.5. Show that Laplace distribution belongs to the semi-α-Laplace family.

4.6.6. Define a geometric exponential distribution similar to the geometric stable distri-bution.

4.6.7. Try to develop a generalized Laplacian model, with characteristic functionϕx(t) =(

11−β2t2

)α

.

4.6.8. Develop the concept min geometric infinite divisibility by replacing addition byminimum in the case of g.i.d.

4.6.9. Develop an autoregressive minification structure by replacing addition by minimumin the standard AR(1) equation.

References

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Pillai, R. N. (1990a). On Mittag-Leffler functions and related distributions,Ann. Inst.Statist. Math.,42, 1, 157–161.

Pillai, R. N. (1990b). Harmonic mixtures and geometric infinite divisibility, J. IndianStatist. Assoc.,28, 87–98.

Pillai, R. N. (1991). Semi–Pareto processes,J. Appl. Prob.,28, 461–465.

Pillai, R. N. and Sandhya, E. (1990). Distributions with complete monotone derivative andgeometric infinite divisibility, Adv. Appl. Prob.,22, 751–754.

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