Convergence Notes

8/6/2019 Convergence Notes

1/6

Convergence of random variablesInprobability theory, there exist several different notions ofconvergence ofrandom

variables. The convergence (in one of the senses presented below) ofsequences of

random variables to some limiting random variable is an important concept in

probability theory, and its applications to statistics and stochastic processes. For

example, if the average ofnuncorrelated random variables Yi, i = 1, ..., n, is given by

then as n goes to infinity,Xn converges in probability (see below) to the common

mean, , of the random variables Yi. This result is known as the weak law of large

numbers. Other forms of convergence are important in other useful theorems,

including the central limit theorem.

Throughout the following, we assume that (Xn) is a sequence of random variables, and

Xis a random variable, and all of them are defined on the sameprobability space (,

F, P).

Contents

1 Convergence in distribution 2 Convergence in probability

o 2.1 Lemmao 2.2 Proof of lemmao 2.3 Proof

3 Almost sure convergence 4 Sure convergence 5 Convergence in mean 6 Implications

7 See also 8 External links 9 References

Convergence in distribution

Suppose thatF1,F2, ... is a sequence ofcumulative distribution functions

corresponding to random variablesX1,X2, ..., and thatFis a distribution function

corresponding to a random variableX. We say that the sequenceXn converges towards

Xin distribution, if


2/6

for every real numbera at whichFis continuous. SinceF(a) = Pr(X a), this means

that the probability that the value ofXis in a given range is very similar to the

probability that the value ofXn is in that range, provided n is large enough.

Convergence in distribution is often denoted by adding the letter over an arrow

indicating convergence:

Small dis also possible, although less common.

Convergence in distribution is the weakest form of convergence, and is sometimes

called weak convergence (main article: weak convergence of measures). It does not,

in general, imply any other mode of convergence. However, convergence in

distribution is implied by all other modes of convergence mentioned in this article,

and hence, it is the most common and often the most useful form of convergence of

random variables. It is the notion of convergence used in the central limit theorem and

the (weak) law of large numbers.

A useful result, which may be employed in conjunction with law of large numbers

and the central limit theorem, is that if a function g: RR is continuous, then

if Xn converges in distribution to X, then so too does g(Xn) converge in

distribution to g(X). (This may be proved using Skorokhod's representation theorem.)This fact could be taken as a definition for the convergence in distribution.

Convergence in distribution is also called convergence in law, since the word "law"

is sometimes used as a synonym of "probability distribution."

Convergence in probability

To say that the sequenceXn converges towardsXin probability means

for every > 0. Formally, pick any > 0 and any > 0. LetPn be the probability that

Xn is outside a tolerance ofX. Then, ifXn converges in probability toXthen there

exists a valueNsuch that, for all nN,Pn is itself less than .

Convergence in probability is often denoted by adding the letter 'P' over an arrow



3/6

Convergence in probability is the notion of convergence used in the weak law of large

numbers. Convergence in probability implies convergence in distribution. To prove it,

it's convenient to prove the following, simple lemma:

Lemma

LetX, Ybe random variables, c a real number and > 0; then

Proof of lemma

since

ProofFor every , due to the preceding lemma, we have:

So, we have

Taking the limit for , we obtain:

But is the cumulative distribution functionFX(a), which is continuous

by hypothesis, that is

and so, taking the limit for , we obtain

Almost sure convergenceTo say that the sequenceXn converges almost surely oralmost everywhere orwith

probability 1 orstrongly towardsXmeans

This means that the values ofXn approach the value ofX, in the sense (see almost

surely) that events for whichXn does not converge toXhave probability 0. Using the

probability space (,F, P) and the concept of the random variable as a function from

to R, this is equivalent to the statement


4/6

Almost sure convergence is often denoted by adding the letters a.s. over an arrow


Almost sure convergence implies convergence in probability, and hence implies

convergence in distribution. It is the notion of convergence used in the strong law of

large numbers.

Sure convergence

To say that the sequence orrandom variables (Xn) defined over the sameprobability

space (i.e., a random process) converges surely oreverywhere orpointwise towards

Xmeans

where is the sample space of the underlyingprobability space over which the

random variables are defined.

This is the notion ofpointwise convergence of sequence functions extended to

sequence ofrandom variables. (Note that random variables themselves are functions).

Sure convergence of a random variable implies all the other kinds of convergence

stated above, but there is no payoff inprobability theory by using sure convergence

compared to using almost sure convergence. The difference between the two only

exists on sets with probability zero. This is why the concept sure convergence of

random variables is very rarely used.

Convergence in mean

We say that the sequenceXn converges in the r-th mean orin theLrnorm towardsX,

ifr 1, E|Xn|r< for all n, and

where the operator E denotes the expected value. Convergence in rth mean tells us

that the expectation of the r-th power of the difference betweenXn andXconverges to

zero.

This type of convergence is often denoted by adding the letterLrover an arrow



5/6

The most important cases of convergence in r-th mean are:

WhenXn converges in r-th mean toXforr= 1, we say thatXn converges inmean toX.

WhenXn converges in r-th mean toXforr= 2, we say thatXn converges inmean square toX.

Convergence in the r-th mean, forr> 0, implies convergence in probability (by

Chebyshev's inequality), while ifr>s 1, convergence in r-th mean implies

convergence ins-th mean. Hence, convergence in mean square implies convergence

in mean.

Implications

The chain of implications between the various notions of convergence are noted in

their respective sections. They are, using the arrow notation

No other implications other than these hold in general, but a number of special cases

do permit the converse implications:

IfXn converges in distribution to a constant c, thenXn converges in probabilityto c.

IfXn converges in probability toX, and if Pr(|Xn| b) = 1 for all n and some b,thenXn converges in rth mean toXfor all r 1. In other words, ifXn

converges in probability toXand all random variablesXn are almost surely

bounded above and below, thenXn converges toXalso in any rth mean.

If for all > 0,

then we say thatXnconverges almost completely, orfast in probability

towardsX. WhenXn converges almost completely towardsXthen it also

converges almost surely toX. In other words, ifXn converges in probability to

Xsufficiently quickly (i.e. the above sequence of tail probabilities is

summable for all > 0), thenXn also converges almost surely toX. This is a

direct implication from the Borel-Cantelli lemma.

IfSn is a sum ofn real independent random variables:

then Sn converges almost surely if and only ifSn converges in probability.

Lvy's convergence theorem gives sufficient conditions for almost sureconvergence to implyL1-convergence:


6/6

Convergence Notes

Documents

Transcript of Convergence Notes