Donsker’s Invariance Principleand Brownian martingales

Updated June 2, 2021

Plan 0

Donsker’s Invariance PrincipleWeak convergence in Wiener spaceTools for verifying tightnessContinuous-time martingalesExamples using Brownian motion

Scaling limit of random walks 1

Brownian motion constructed as a Cpr0, 8qq-valued r.v.Original motivation: scaling limit of random walksLet Z1, Z2, . . . be i.i.d. R-valued r.v.’s and set

@n P N : Xn :“nÿ

k“1

Zk

Use these to construct an element of Cpr0, 8qq via

@t P r0, 8q : Ypnqt :“ 1?

n

´Xtntu ` pnt ´ tntuqXtntu`1

¯

Consequences of CLT 2

If EpZ1q “ 0 and EpZ21q “ s2, then by CLT:

@t • 0 : Ypnqt

law›ÑnÑ8 N p0, s2tq

Note: sBtlaw“ N p0, s2tq.

Mutlivariate CLT even gives convergence in the sense of finitedimensional distributions:

@0 § t1 † ¨ ¨ ¨ † tk : pYpnqt1

, . . . , Ypnqtk

˘ law›ÑnÑ8 psBt1 , . . . , sBtkq

where B is the SBM.

Q: Convergence of the law of t fiÑ Ypnqt on Cpr0, 8qq?

Donsker’s Invariance Principle 3

Theorem (Donsker 1951)For Xn :“ Z1 ` ¨ ¨ ¨ ` Zn with tZkuk•1 i.i.d. satisfying EpZ1q “ 0and EpZ2

1q “ 1, as n Ñ 8 the law of

Ypnqt :“ 1?

n

´Xtntu ` pnt ´ tntuqXtntu`1

¯

on pCpr0, 8qq,BpCpr0, 8qqqq converges weakly to Wiener measure.

Precise meaning of convergence? Define

@A P BpCpr0, 8qqq : PpnqpAq :“ PpYpnq P Aq

Donsker’s Theorem says: Ppnq wÑ PW as n Ñ 8.

Added value 4

Q: Why is this more than just conv. of finite-dim. distributions?

CorollaryFor above setting,

1?n

max1§k§n

Xklaw݄

nÑ8 max0§t§1

Bt

Corollary

Given a ° 0, set Tpnqa :“ inf

k • 0 : Xk • a

(. Then

1n

Tpnqa?

nlaw݄

nÑ8 inftt • 0 : Bt • au

Proof of Corollaries 5

Proof of Corollaries 6

How to prove Donsker’s theorem? 7

Theorem

Let tPpnqun•1 and P be probability measures on the Wiener spacepCpr0, 8qq,BpCpr0, 8qqqq. Then

Ppnq w›ÑnÑ8 P

is equivalent to the conjunction of(1) Ppnq Ñ P in the sense of finite-dimensional distributions(2) tPpnqun•1 is tight

Recall: tPpnqun•1 on pX ,BpX qq is tight if

@e ° 0 DK Ñ X compact : lim supnÑ8

PpnqpX r Kq † e

Proof of Theorem 8

Verifying tightness 9

Arzela-Ascoli Theorem: A set K Ñ Cpr0, 8qq is compact if andonly if for each M • 1 the set

w|r0,Ms : w P K

(

is closed in Cpr0, Msq, pointwise bounded and equicontinuous.

Equicontinuity hard: needs truncation (to increase availablemoments) & Kolmogorov inequality (to control oscillation ofpaths over intervals)

We will prove Donsker’s Theorem via Martingale FunctionalCentral Limit Theorem (to be discussed next time)

Continuous-time martingales 10

DefinitionAn R-valued process tXtut•0 is a martingale with respect tofiltration tFtut•0 if(1) @t • 0 : Xt is Ft-measurable with Xt P L1, and(2) @t, s • 0 : EpXt`s|Ftq “ Xt a.s.

Note:submartingale if EpXt`s|Ftq • Xt, supermartingale if “§”continuous/cadlag (sub/super)martingale if every samplepath t fiÑ Xt is continuous/cadlag

Fact (275D): regularity assumptions sometimes superfluous;sub/supermartingales admit cadlag versions

Brownian martingales 11

The following are continuous martingales

Bt, B2t ´ t, B3

t ´ 3tBt, B4t ´ 6tB2

t ` 3t2, . . .

These are all generated by continuous martingale

Mt :“ elBt´ l22 t

Indeed,

elBt´ l22 t “ 1 ` lBt ` l2

2pB2

t ´ tq

` l3

6pB3

t ´ 3tBtq ` l4

24pB4

t ´ 6tB2t ` 3t2q ` . . .

Basic facts derived via discrete-time martingales 12

LemmaX martingale ñ tXt : t § au UI for all a P p0, 8q

LemmaX cadlag martingale ^ T stopping time ñ tXT^tut•0 martingale

18

TO BE CONTINUED . . .