Donsker s Theorem. Pierre Yves Gaudreau Lamarre. August 2012

Transcription

1 Dosker s heorem Pierre Yves Gaudreau Lamarre August 2012 Abstract I this paper we provide a detailed proof of Dosker s heorem, icludig a review of the majority of the results o which the theorem is based, ad we desig algorithms that provide experimetal evidece for the Classical Cetral Limit heorem ad that allow us to observe the effects of Dosker s heorem. Ackowledgmets. I would like to thak the Uiversity of Ottawa Work tudy Program for makig this work term possible ad Professor Raluca Bala for acceptig to hire me as her assistat for the summer.

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3 Cotets 1 Weak Covergece i Metric paces Defiitio Criteria for Weak Covergece Prohorov s heorem Covergece i Distributio Notes o Other Modes of Covergece paces of Fuctios Cotiuous Fuctios o [0,1] Dosker s heorem Numerical Verificatios Classical Cetral Limit heorem Dosker s heorem A Appedix - heoretical Backgroud 46 A.1 Measure heory A.2 Itegratio A.3 Lebesgue Measure A.4 Probability Idex 56

4 1 Weak Covergece i Metric paces I this sectio we itroduce the cocept of weak covergece i metric spaces, otherwise kow as covergece i distributio i particular cotexts, which will be fudametal to the study of cetral limit theorems. Let, ρ) be a arbitrary metric space. Give x ad ε > 0 we use Bx, ε) to deote the set {y : ρx, y) < ε}, ad to deote the Borel σ-field o. Give A, we use the otatio A for the boudary of A, A for the iterior of A, A for the closure of A ad I A for the idicator fuctio of A. 1.1 Defiitio Defiitio 1.1 Let, ρ) be a metric space ad P, {P } N be probability measures o, ). w We say that the sequece {P } N coverges weakly to P ad write P P if, for every fuctio f : R that is bouded ad cotiuous, we have f dp f dp 1. o esure that the cocept of weak covergece is well defied, it is ecessary to verify that w the limit for P P is uique. Defiitio 1.2 Let, τ) be a arbitrary topological space. A probability measure P o, ) is regular if, for every A ad every ε > 0, there exists a ope set G ad a closed set F of such that F A G ad PG \ F ) < ε. Propositio 1.3 [2] heorem 1.1) Let, ρ) be a metric space. Every probability measure P o, ) regular. Propositio 1.4 Let, ρ) be a metric space ad P, P be probability measures o, ). If PF ) = P F ) for every closed set F, the P = P. Proof Let A ad ε > 0 be arbitrary. By Propositio 1.3, there exists a ope set G ad a closed set F such that F A G ad PG \ F ) < ε. As probability measures are mootoe, A G implies that PA) PG), ad F A implies that P A) P F ). We the have PA) P A) PG) P F ) = PF G \ F )) P F ) sice F G) = PF ) + PG \ F ) P F ) F ad G \ F are disjoit) = PG \ F ) F is closed) < ε. akig the limit as ε 0, we obtai the equality PA) = P A). Lemma 1.5 [2] heorem 1.2) Let, ρ) be a metric space ad P, P be probability measures o, ). If f dp = f dp for every bouded ad uiformly cotiuous fuctio f : R, the P = P. Proof Let F be a arbitrary closed set. For every ε > 0, defie F ε := {x : ρx, F ) < ε} ad f ε : R as 1 Give a sequece {x } N, we write x x for lim x = x to alleviate otatio. 4

5 1 ρx, F ) if x F f ε x) = ε ε 0 otherwise Note that f ε is bouded by 1, uiformly cotiuous o ad for every x, we have that I F x) f ε x) I Fε x). hus, PF ) = I F dp f ε dp by mootoicity, I F f ε ) = f ε dp by hypothesis, f ε bouded ad uiformly cotiuous) I Fε dp by mootoicity, f ε I Fε ) = P F ε ). As the above holds for every ε > 0, we take a limit as ε 0 ad coclude PF ) P F ). Usig a similar argumet, we could prove P F ) PF ), which would imply PF ) = P F ). he result the follows by Propositio 1.4. With these results, we ca obtai the followig theorem, which establishes that a sigle sequece of probability measures caot coverge weakly to two differet probability measures. heorem 1.6 Let, ρ) be a metric space ad P, P, {P } N be probability measures o, ). If P w P ad P w P, the P = P. Proof Let f : R be a arbitrary bouded ad cotiuous fuctio. he, by the defiitio of weak covergece f dp = lim f dp = f dp. ice every uiformly cotiuous fuctio is cotiuous, we ca apply Lemma 1.5 to obtai the equality P = P. 1.2 Criteria for Weak Covergece It is ofte impractical to demostrate weak covergece for a sequece of probability measures {P } N by takig a limit as of itegrals with respect to dp for a arbitrary bouded cotiuous fuctio. We will ow see that there are various criteria that imply or are equivalet to weak covergece that, i some cases, ca be easier to work with. Defiitio 1.7 Let, ρ) be a metric space ad P be a probability measure o, ). A set A is called a P-cotiuity set if P A) = 0. heorem 1.8 Portmateau heorem) [2] heorem 2.1) Let, ρ) be a metric space ad P, {P } N be probability measures o, ). he followig coditios are equivalet. w i. P P ii. f dp f dp for every bouded, uiformly cotiuous fuctio f : R iii. lim sup P F ) PF ) for every closed set F iv. lim if P G) PG) for every ope set G v. P A) PA) for every P-cotiuity set A 5

6 Proof i. = ii. his is a direct cosequece of the defiitio of weak covergece, sice every uiformly cotiuous fuctio is cotiuous. ii. = iii. Let F be a arbitrary closed set. Give ay ε > 0, let F ε ad f ε : R be defied as i the proof of Lemma 1.5. We the have lim sup P F ) = lim sup I F dp lim sup f ε dp by mootoicity, I F f ε ) = f ε dp by ii, f ε is bouded ad uiformly cotiuous) I Fε dp by mootoicity, f ε I Fε ) = PF ε ). akig the limit ε 0, we have lim sup P F ) PF ). iii. iv. uppose that iii. holds. Let G be a arbitrary ope set. ice G is ope, G c is closed, ad therefore, by hypothesis ad the properties of iferior ad superior limits, we obtai: lim if P G) 1) = lim if 1 P G)) = lim if G c ) = lim sup P G c ) PG c ) by iii, G c is closed) = PG) 1, which proves iv. A similar argumet shows that iv. implies iii. iv. = v. Let A be a arbitrary P-cotiuity set of. We first otice that PA ) = PA \ A ) A ) sice A A ) = P A A ) = P A) + PA ) iterior ad boudary disjoit) = PA ). A is a P-cotiuity set) Moreover, we have A A A, which implies that PA ) PA) PA ) by mootoicity, ad therefore, PA ) = PA) = PA ). By hypothesis ad equivalece of iii. ad iv, ad thus, PA) = PA ) lim sup P A ) by iii. sice A is closed) lim sup P A ) by mootoicity sice A A ) lim if A ) PA ) by iv. sice A is ope) = PA) PA) = lim sup P A ) = lim if P A ). 6

7 Cosequetly, by the equality PA) = PA ) = PA ), we have which implies that P A) PA). PA) = lim sup P A) = lim if P A), v. = i. Let f : R be a arbitrary cotiuous fuctio such that 0 < fx) < 1 for every x. For every t 0, 1), defie the set [f > t] = {x : fx) > t}. Let x [f > t] \ [f > t]. We will prove fx) = t. he fact that x [f > t] implies that fx) t, ad x [f > t] implies that there exists a sequece {x } N [f > t] such that x x. uppose fx) = k < t. he, there exists some ε > 0, amely, ε = t k, such that, for every N, fx ) fx) = fx ) k > t k sice N : x [f > t]) = ε. he above implies that fx ) fx), which cotradicts the fact that f is cotiuous. hus, it is ecessary that fx) = t. From here, it follows that [f > t] {x : fx) t} = [f t]. 1) Let x [f > t], ad k := fx) > t. ice f is cotiuous, for ε = k t > 0, there exists δ > 0 such that for ay y Bx, δ), we have fx) fy) = k fy) < ε = k t. his implies that fy) > t for every y Bx, δ). hus, Bx, δ) [f > t], which implies that [f > t] is ope sice x was arbitrary. We the have [f > t] = [f > t]. 2) We ca ow show the followig with regards to the boudary of [f > t]: [f > t] = [f > t] \ [f > t] [f t] \ [f > t] by 1)) = [f t] \ [f > t] by 2)) = {x : fx) = t} By Corollary A.17, f is a radom variable, thus, for every probability measure Q o, it has a distributio Qf 1 with associated distributio fuctio F Qf. herefore, Q [f > t]) Q{x : fx) = t}) = F Qf t) lim x t F Qf x) which, by Propositio A.45, is ozero for at most coutably may t. Applyig this to P, we have that the sets of the form [f > t] are P-cotiuity sets except for at most coutably may t, ad cosequetly, by coditio v. P [f > t]) P[f > t]) except for at most coutably may t. We the have lim f dp = lim = lim = = 0,1) [0, ) 0,1) P [f > t]) dt by Propositio A.56) P [f > t]) dt sice 0 < fx) < 1) P[f > t]) dt f dp. 7 by the bouded covergece heorem) by Propositio A.56)

8 o see how the above equality geeralizes to every bouded cotiuous fuctio, we otice that for every cotiuous fuctio f : R that is bouded i.e. there exists some M R such that M < fx) < M for every x ) the fuctio g : R defied as gx) = fx)+m M is such that 0 < gx) < 1 for every x. he coclusio follows by the liearity of itegrals. Let, ρ) ad, ρ ) be metric spaces, P be a probability measure o, ) ad h : be / -measurable. he 2, P h 1 is a probability measure o, ) 3. Give a sequece of probability measures {P } N o, ), the followig result gives ecessary coditios o h for P w P to imply P h 1 w P h 1. heorem 1.9 Cotiuous Mappig heorem) [2] heorem 2.7) Let, ρ) ad, ρ ) be metric spaces, P, {P } N be probability measures o, ) ad h : be a arbitrary / -measurable fuctio. Let D h be the set of poits at which h is discotiuous. If PD h ) = 0 ad P w P, the P h 1 w Ph 1. Proof Let F be a arbitrary closed set. If x is a poit i h 1 F )) we kow there exists a sequece {x } N h 1 F ) such that x x. uppose x D h. he, h is cotiuous at x, which implies that hx ) hx), ad sice hx i ) F for every i 0, hx) is a limit poit of F, hece, hx) F. Cosequetly, x D c h implies that x h 1 F ). his proves that D c h h 1 F )) h 1 F ). We the have lim sup P h 1 F ) = lim sup P h 1 F )) lim sup P h 1 F )) ) sice h 1 F ) h 1 F )) ) P h 1 F )) ) Portmateau theorem, h 1 F )) is closed) = P D c h h 1 F )) ) Propositio A.37 b. sice PD c h) = 1) Ph 1 F )) sice D c h h 1 F )) h 1 F )) = Ph 1 F ) F is closed, hece equal to its closure). By the Portmateau theorem, it follows that P h 1 w Ph 1. I the above result, o metio was made about coditios o h for D h to be measurable. We see that this is always the case. Let, ρ) ad, ρ ) be metric spaces ad h : be a arbitrary measurable fuctio. Give ε > 0 ad δ > 0, defie the set A ε,δ = {x : y, z such that ρx, y) < δ, ρx, z) < δ ad ρ hy), hz)) ε} Lemma 1.10 For every ε > 0 ad δ > 0, A ε,δ is ope. Proof Let ε > 0 ad δ > 0 be arbitrary. Give x A ε,δ, there exists y, z such that ρx, y) < δ, ρx, z) < δ ad ρ hy), hz)) ε. Let η = mi{δ ρy, x); δ ρx, z)}. We show that Bx, η) A ε,δ. Let x be i Bx, η). By the triagle iequality, we have ad similarly, 2 Where deotes fuctio compositio 3 ee Propositio A.15 ad remark A.43 ρx, y) ρx, x) + ρx, y) < η + ρx, y) δ ρx, y) + ρx, y) = δ, ρx, z) ρx, x) + ρx, z) < η + ρx, z) δ ρx, z) + ρx, z) = δ. 8

9 Hece, there exists y, z such that ρx, y) < δ, ρx, z) < δ ad ρ hy), hz)) ε, which implies that x A ε,δ. As x Bx, η) was arbitrary, we coclude Bx, η) A ε,δ. ice x, ε ad δ were also arbitrary, it follows that, for every ε > 0 ad δ > 0, for every x A ε,δ, there exists a ope ball Bx, η) cotaied i A ε,δ, from which the result follows. Propositio 1.11 For every measurable fuctio h :, D h is a measurable set. Proof By the defiitio of cotiuity o metric spaces, h is discotiuous at x if ad oly if there exists some ε > 0 such that for every δ > 0, x A ε,δ. Hece D h = ) A ε,δ. ε>0 ice the positive ratioal umbers are dese i the positive real umbers, we see that D h = ) A ε,δ =, ε>0 δ>0 δ>0 ε Q + δ Q + A ε,δ where Q + = {x Q : x > 0}. From Lemma 1.10, we have that every A ε,δ is ope, so obviously a elemet of. It the follows from Propositio A.4 ad the defiitio of σ-fields that D h. heorem 1.12 [2] heorem 2.6) Let, ρ) be a metric space ad P, {P } N be probability measures o, ). P w P if ad oly if for every subsequece {Pi } i N, there exists a further subsequece {P ij } j N such that P ij w P. w w Proof he fact that P P implies that Pij P is obvious from the defiitio of weak covergece. w We ow show the other implicatio by cotrapositio. uppose P P. Cosequetly, there exists a bouded ad cotiuous fuctio f : R such that f dp f dp I other words, there exists ε > 0 such that, for every N N, there is some N such that f dp f dp > ε We ca therefore fid a icreasig sequece of idices 1 < 2 <... such that the subsequece {P i } i N has the followig property f dp i f dp > ε for every i N. Obviously o further subsequece of {P i } i N coverges weakly to P. heorem 1.13 Give a iteger k 2, let 1, 1 ),..., k, k ) be measurable metric spaces 4, 1... k, 1... k ) be the product space o 1,..., k ad for every i k, let P i, {P i } N be probability measures o i, i ). If 1... is separable, the the sequece of product measures {P 1... P k } N coverges weakly to the product measure P 1... P k if ad oly if for every i k, P i w P i. 4 For every i k, i deotes the Borel σ-field o i 9

10 Proof We proceed by iductio. he case k = 2 follows from [2] heorem 2.8 ii). uppose the result holds for some k = m 2. We prove it holds for k = m + 1. uppose 1... m+1 = 1... m ) m+1 is separable. From the case k = 2, we kow that P 1... P m ) P m+1 w P 1... P m ) P m+1 if ad oly if P 1... P m w P 1... P m, ad P m+1 w P m+1. 3) From [2] Appedix M10, a product of two metric spaces 1 2 is separable if ad oly if 1 ad 2 are separable. Cosequetly, the fact that 1... m ) m+1 is separable implies that 1... m is separable. ice the result holds for k = m, we the have that P 1... P m w P 1... P m if ad oly if for every i m, P i w P i. he result follows by combiig the last remark with 3). 1.3 Prohorov s heorem Defiitio 1.14 A set K of probability measures o, ) is relatively compact if for every sequece of probability measures {P } N K, there exists a subsequece of {P } that coverges weakly to some probability measure P o, ). Defiitio 1.15 A probability measure P o, ) is tight if, for every ε > 0, there exists a compact set K such that PK) > 1 ε. imilarly, a set K of probability measures o, ) is tight if for every ε > 0, there exists a compact set K such that PK) > 1 ε for every P K. heorem 1.16 Prohorov s heorem Direct Part) [2] heorem 5.1) Let K be a collectio of probability measures o, ). If K is tight, the K is relatively compact. heorem 1.17 Prohorov s heorem Coverse Part) [2] heorem 5.2) Let K be a collectio of probability measures o, ). If, ρ) is a separable ad complete metric space ad K is relatively compact, the K is tight. Proof Let {G } N be a sequece of ope sets such that G i G j if i j ad G. We first prove that, for every ε > 0, there exists N N such that for every P K, if N, the PG ) > 1 ε. 4) o show this, suppose by cotradictio that there exists some ε > 0 such that, for every N, there exists P K such that P G ) 1 ε. ice K is relatively compact, there exists a subsequece {P i } i N of {P } that coverges weakly to some probability measure P o, ). ice every G is ope, it follows from the Portmateau theorem that, for every N PG ) lim if i P i G ) lim if P i G i ) by mootoicity, G i G j for i j) i 1 ε, which cotradicts the fact that G by Propositio A.10 c. ice is separable, for every k > 0, there exists a coutable set {A k, } N of ope balls of radius 1/k that covers. Applyig 4), we kow that for ay k > 0 ad every ε > 0, there exists k N such that, for ay P K, if k, the P A k,i > 1 ε 2 k. 5) Defie the set i 10

11 K := k=1 i k A k,i For every k > 0, the set i k A k,i is obviously totally bouded, ad sice is complete, i k A k,i is also compact 5. ice every itersectio of compact sets is compact 6, K is compact. Moreover, for every probability measure P K PK) = 1 PK c ) = 1 P k=1 = 1 P 1 = 1 = 1 + > 1 + = 1 ε k=1 P k=1 i k A k,i i k A k,i i k A k,i 1 P k=1 P k=1 k=1 c c c i k A k,i i k A k,i [ 1 ε ) ] 2 k 1 k=1 1 2 k 1. DeMorga s law) coutable subadditivity) by 5)) = 1 ε geometric series). Remark 1.18 A obvious corollary to the coverse part of Prohorov s heorem is that, if, ρ) is a complete ad separable metric space, the every probability measure o, ) is tight. Ideed, for every probability measure P, the sigleto {P} is relatively compact, sice its oly sequece P, P,... coverges weakly to P. 1.4 Covergece i Distributio We ow look at the particular case of weak covergece for the distributios of radom elemets. Defiitio 1.19 Let Ω, F, P), {Ω, F, P )} N be probability spaces,, ρ) be a metric space ad X : Ω, {X : Ω } N be radom elemets with respective distributios µ X, {µ X } N. We say {X } N coverges i distributio to X ad write X d X if µx w µx. We ca rewrite the Portmateau ad cotiuous mappig theorems i the preset cotext ad directly obtai the followig modificatios of the results. Defiitio 1.20 Let Ω, F, P) be a probability space,, ρ) be a metric space ad X : Ω be a radom elemet with distributio µ X. A set A is called a X-cotiuity set if µ X A) = 0. 5 ee [4] heorem 92 6 ee [5] Corollaire

12 heorem 1.21 Portmateau heorem) Let Ω, F, P), {Ω, F, P )} N be probability spaces,, ρ) be a metric space ad X : Ω, {X : Ω } N be radom elemets with respective distributios µ X, {µ X } N. he followig coditios are equivalet i. X d X ii. E[fX )] E[fX)] for every fuctio f : R that is bouded ad uiformly cotiuous iii. lim sup µ X F ) µ X F ) for every closed set F iv. lim if µ X G) µ X G) for every ope set G v. µ X A) µ X A) for every X-cotiuity set A Let Ω, F, P) be a probability space ad, ) ad, ) be measurable spaces. By heorem A.14, we kow that, give a radom elemet X : Ω ad a / -measurable mappig h :, the compositio hx) : Ω is a radom elemet. heorem 1.22 Cotiuous Mappig heorem) Let Ω, F, P), {Ω, F, P )} N be probability spaces,, ) ad, ) be measurable spaces, X : Ω, {X : Ω } N be radom elemets, h : be a / -measurable mappig ad D h be the set of poits at d which h is discotiuous. If X X ad PX Dh ) = 0 7, the hx ) d hx). Defiitio 1.23 Let Ω, F, P) ad Ω, F, P ) be probability spaces,, ) be a measurable metric space ad X : Ω ad Y : Ω be radom elemets. We say X ad Y are equal i distributio ad write X = d Y if µ X = µ Y, where µ X ad µ Y are the distributios of X ad Y respectively. Propositio 1.24 Let Ω, F, P), {Ω, F, P )} N ad {Ω, F, P )} N be probability spaces,, ) be a measurable metric space ad X : Ω, {X : Ω } N ad {Y : Ω } N be radom elemets. If X d X ad X d = Y for every N, the Y d X. Proof Let f : R be a arbitrary bouded ad cotiuous fuctio. he, d lim f dµ Y = lim f dµ X sice X = Y N) = f dµ X, which proves Y d X. 1.5 Notes o Other Modes of Covergece Defiitio 1.25 Let Ω, F, P) be a probability space,, ρ) be a metric space ad X, {X } N be p radom elemets o Ω. We say {X } N coverges i probability to X ad write X X if, for every ε > 0 PρX, X) < ε) 1. Propositio 1.26 [2] heorem 3.1) Let {X } N ad {Y } N be sequeces of radom variables o a probability space Ω, F, P) takig values i, ). If X X ad ρx, Y ) p 0, d d the Y X. 7 ee Defiitio A.42 12

13 Proof For a arbitrary closed set F ad ε > 0, let F ε = {x : ρx, F ) ε}. For every N, if ω Ω is such that Y ω) F ad X ω) F ε, the ρy ω), X ω)) > ε. herefore, {ω Ω : Y ω) F } \ {ω Ω : X ω) F ε } {ω Ω : ρy ω), X ω)) > ε}. By mootoicity ad Propositio A.10 e. we the have PY F ) PρY, X ) > ε) + PX F ε ). By hypothesis ad the Portmateau heorem, sice F ε is closed, we have Lettig ε 0 we have lim sup PY F ) 0 + lim sup PX F ε ) PX F ε ) lim sup PY F ) PX F ) for every closed set F, which proves Y w X by the Portmateau heorem. heorem 1.27 lutsky s heorem)[10] heorem 6.1) Let {X } N, {Y } N ad X be d p radom variables o a probability space Ω, F, P). If X X ad Y c, where c R, the d a. X + Y X + c d b. X Y cx 13

14 2 paces of Fuctios 2.1 Cotiuous Fuctios o [0,1] Let C be the set of all cotiuous fuctios x : [0, 1] R with respect to the stadard Euclidea metric dt 1, t 2 ) = t 1 t 2. As C is a real vector space, we ca defie a orm o its elemets from which we ca obtai a metric. Give x C, let x := sup xt) t [0,1] he, C together with the distace ρx 1, x 2 ) = x 1 x 2 is a metric space. Note that, throughout this sectio, we will deote by C the Borel σ-field o C, ad by B k the Borel σ-field o R k for every k N. Propositio 2.1 he space C, ρ) is complete. Proof Let {x } N be a arbitrary Cauchy sequece of fuctios i C. hus, for every ε > 0, there exists N N such that if, m N, the x m x < ε. I particular, for every t [0, 1], x t) x m t) sup x t) x m t) = x x m < ε, t [0,1] which implies that {x t)} N is a Cauchy sequece of real umbers. ice R with the stadard euclidea metric is a complete metric space, we kow that for every t [0, 1], the sequece {x t)} has a limit x t R. Defie the fuctio x : [0, 1] R as xt) = x t for every t [0, 1]. We show that x x i the metric ad that x C. ice {x } N is a Cauchy sequece, for every ε > 0, there exists N N such that if, m N, the x x m < ε. hus, for every t [0, 1], we have x t) x m t) < ε, which implies that x t) xt) = lim m x t) x m t) ε. ice this is true for every t [0, 1], we have x x ε, which proves x x i C. he fact that x C follows from a classical result i fuctioal aalysis 8. Cosequetly, every Cauchy sequece of fuctios i C has a limit i C, which proves C, ) is complete. Propositio 2.2 he space C, ρ) is separable. Proof For every k N, let Q k C be the set of polyomials with ratioal coefficiets of degree k. he, for every k N, we have Q k Q k where deotes cardiality), which implies that Q := k N Q k is a coutable set, sice it is a coutable uio of coutable sets. Let pt) = a 0 + a 1 t a k t k be a polyomial of degree k with real coefficiets. ice Q is dese i R, for every ε > 0 ad i k, there exists b i Q such that a i b i <. If we defie qt) = b 0 + b 1 t b k t k, the for every t [0, 1] we have k pt) qt) = a i b i )t i < i=0 k a i b i i=0 k i=0 ε 2k + 1) = ε 2. 8 ee [5] heorem 5.10 Uiform limit of cotiuous fuctios is cotiuous) ε 2k+1) 14

15 ice the above holds for every t [0, 1] ad k N, we have that, for every polyomial with real coefficiets p C ad ε > 0, there exists a polyomial q Q with ratioal coefficiets such that ρp, q) = p q ε/2. Furthermore, we kow from the Weierstrass approximatio heorem 9 that, give ay x C, for every ε > 0, there exists a polyomial p C with real coefficiets such that ρx, p) < ε/2. Hece, for every x C ad ε > 0, there exists a polyomial p C with real coefficiets ad a polyomial q Q with ratioal coefficiets such that ρx, q) ρx, p) + ρp, q) ε. his proves that Q that is dese i C. Cosequetly, C has a coutable dese set, which makes it separable. Corollary 2.3 Every probability measure o C, ρ) is tight. Proof his follows from Propositio 2.1, Propositio 2.2 ad Remark ome useful criteria for covergece of probability measures o C, C) ivolves mappigs that project fuctios of C i Euclidea space. Defiitio 2.4 For every iteger k 1, defie the set U k = {t 1,..., t k ) [0, 1] k : t 1 <... < t k }. Defiitio 2.5 Give k 1 ad = t 1,..., t k ) U k, defie the projectio mappig, deoted π : C R k, as for every x C. π x) := xt 1 ),..., xt k )) Propositio 2.6 For every iteger k 1 ad = t 1,..., t k ) U k, the projectio π is cotiuous everywhere o C with respect to the euclidea l 2 metric hece C/B k -measurable 10 ). Proof For every x 1 C ad ε > 0, there exists δ > 0, amely δ = ε k, such that if x 2 Bx 1, δ), the π x 1 ) π x 2 ) l2 = k ) 1/2 x 1 t i ) x 2 t i ) 2 k sup t [0,1] < k δ = ε. x 1 t) x 2 t) 2 ) 1/2 Defiitio 2.7 A set A C is called fiite-dimesioal if there exists U k for some k 1 ad H B k such that π 1 H) = A. Let C f be the class of all fiite-dimesioal sets. Remark 2.8 We otice that C f C. o see this, let A C f. he, there exists U k with k 1 ad H B k such that π 1 H) = A. From Propositio 2.6, we kow π is C/B k - measurable, which by defiitio implies that A C. Propositio 2.9 [2] p.12) Let P, P be probability measures o C, C). If PA) = P A) for every A C f, the P = P. 9 ee [5] heorem ee Corollary A.17 15

16 Proof We first prove that C f is a π-system. Let A, B C f be arbitrary. he, we kow there exists H A B k ad H B B l, where k, l 1, ad = t 1,..., t k ) U k, = s 1,..., s l ) U l such that A = π 1 H A) ad B = π 1 H B). If we set U = t 1,..., t k, s 1,..., s l ), we the have A B = π 1 H A) π 1 H B) = {x C : xt 1 ),..., xt )) H A ad xs 1 ),..., xs l )) H B } = {x C : xt 1 ),..., xt ), xs 1 ),..., xs l )) H A H B } = π 1 U H A H B ). As B k+l = σ{a B : A B k ad B B l }) 11, we have that H A H B B k+l, which implies that A B C f. ice A ad B were arbitrary, C f is a π-system. We ow prove σc f ) = C. From Remark 2.8, we have that σc f ) C, sice σc f ) is the itersectio of every σ-field that cotais C f. We ow show that Bx, ε) σc F ) for ay x C, ε > 0. Give arbitrary x C ad ε > 0, from the defiitio of the orm ρ o C, we have Bx, ε) = {x C : xt) x t) < ε}. ice Q is dese i R, Bx, ε) = t [0,1] t Q [0,1] {x C : xt) x t) < ε}. We otice that, for ay t Q [0, 1], πt 1 Bxt), ε)) C f, sice Bxt), ε) B. Also, by defiitio, ) πt 1 Bxt), ε) = {x C : π t x ) Bxt), ε)} = {x C : x t) Bxt), ε)} = {x C : xt) x t) < ε}. As σ-fields are closed uder coutable itersectios 12, Bx, ε) σc f ), hece, σc f ) cotais every ope ball i C. Furthermore, sice C is separable by Propositio 2.2, every ope set of C is the uio of at most coutably may ope balls, hece, every ope set of C is i σc f ). ice C is the itersectio of every σ-field cotaiig all the ope sets of C, we have C σc f ). ice C σc f ) ad σc f ) C, we coclude σc f ) = C. P = P the follows from heorem A.12. We kow from Propositio 2.6 that every projectio mappig π with U k is C/B k - measurable, ad thus, for every probability measure P o C, C), we ca defie a probability measure P π 1 o R k, B k ) 13. heorem 2.10 [2] Example 5.1) Let P, {P } N be probability measures o C, C). If for every k 1 ad every U k we have that P π 1 w P π 1 ad {P } N is relatively compact, w the P P. Proof ice the set {P } N is relatively compact, every sequece of elemets i that set cotais a weakly coverget subsequece. I particular, for every subsequece {P i } i N of w {P }, there exists a further subsequece {P ij } j N such that P ij P for some probability measure P o C, C). For every k 1 ad U k, it follows from the Cotiuous Mappig heorem ad Propositio 2.6 that P ij π 1 w P π 1. By heorem 1.6, we the have that P π 1 = P π 1 for every projectio mappig π, as P π 1 w P π 1 obviously implies that P ij π 1 w P π 1. Give a arbitrary A C f, we kow there exists U k for some k 1 ad H B k such that A = π 1 H). We the have 11 ee [6] Propositio ee Propositio A.4 13 ee Propositio A.15 ad remark A.43 16

17 PA) = P π 1 H) = P π 1 H) = P A). It the follows from Propositio 2.9 that P = P. We have just show that, for every subsequece {P i } i N of {P }, there exists a further w subsequece {P ij } j N that coverges weakly to P, which, by heorem 1.12, implies that P P. Corollary 2.11 [2] heorem 7.1) Let P, {P } N be probability measures o C, C). If {P } N is tight ad P π 1 w P π 1 for every iteger k 1 ad every U k w, the P P. Proof his is a cosequece of heorem 2.10 ad the direct part of Prohorov s heorem. he results i this sectio up to this poit give useful criteria to show weak covergece of probability measures o the space C. However, attemptig to prove a sequece of probability measures is tight usig the defiitio of tightess aloe ca sometimes be problematic. We will ow go over some coditios that are equivalet to tightess i C, C) ad are much easier to work with. Defiitio 2.12 Let x C. Give δ > 0, defie the modulus of cotiuity of x for δ, deoted wx, δ), as wx, δ) := sup t,s [0,1] t s δ xs) xt) Remark 2.13 A direct cosequece of the defiitio of covergece for sequeces ad uiform cotiuity is that a fuctio x C is uiformly cotiuous if ad oly if wx, δ) 0 as δ 0. Propositio 2.14 For a fixed δ > 0, w, δ) : C R is cotiuous o C. Proof Let x C be arbitrary ad t, s [0, 1] be such that t s < δ. For every x C, we have xt) xs) = xt) xs) + x t) x t) + x s) x s) xt) x t) + x s) xs) + x t) x s) 2 x x + x t) x s) akig a supremum o both sides of the iequality with respect to all couples t, s [0, 1] such that t s < δ, we obtai wx, δ) 2 x x + wx, δ) 6) Usig the same argumet but iterchagig x with x gives us wx, δ) 2 x x + wx, δ) 7) Depedig o which of wx, δ) or wx, δ) is smaller, wx, δ) wx, δ) is either equal to wx, δ) wx, δ) or wx, δ) wx, δ). I both cases, we coclude from 6) ad 7) that, for ay x, x C, wx, δ) wx, δ) 2 x x. herefore, for every x C ad ε > 0, there exists η > 0, amely, η = ε/2, such that if x Bx, η), the wx, δ) wx, δ) 2ρx, x ) < 2η = ε his proves that w, δ) is cotiuous at x for every x C. 17

18 Defiitio 2.15 A set F of real-valued fuctios that are defied o the same domai D R is uiformly bouded if there exists M > 0 such that, for every x F ad t D, we have xt) < M, ad uiformly equicotiuous if for every ε > 0, there exists δ > 0 such that, if t, s D satisfy t s < δ, the xt) xs) < ε for every x F. heorem 2.16 Arzela-Ascoli heorem) [7] heorem 7.6.1) Let {x } N be a sequece of real-valued fuctios defied o a compact iterval [a, b] R. If the set {x } N is uiformly bouded ad uiformly equicotiuous, the there exists a subsequece of {x } that coverges uiformly to some fuctio x : [a, b] R. he hypotheses of the above theorem ca be expressed i a way that is better suited i our cotext. First, every fuctio i C is obviously a real-valued fuctio defied o a compact iterval of R, so the result applies for every sequece of fuctios i C. ecodly, uiform covergece for fuctios is covergece i the l orm, which is the metric ρ we are usig, so we will simply express uiform covergece of a subsequece as covergece i the space C, ρ). As for uiform equicotiuity ad uiform boudedess, we have the followig lemma. Lemma 2.17 [2] proof of heorem 7.2) uppose A C is such that i. sup x0) = M < x A ) ii. lim δ 0 sup wx, δ) x A = 0 he A is uiformly bouded ad uiformly equicotiuous. Proof Give ay x A, k N ad t [0, 1], we have k ) )) it i 1)t xt) = x0) + x x k k k ) ) x0) + it i 1)t x x 8) k k From ii. we kow that for every fiite > 0, there exists some k N large eough such that if δ 1/k, the sup wx, δ) <. x A Combiig this with i. ad takig a supremum with respect all x A o both sides of the iequality 8) above, we obtai sup xt) < M + k < x A ad sice this holds for every t [0, 1], A is uiformly bouded. Coditio ii. implies that, for every ε > 0, there exists η > 0 such that if δ η, sup wx, δ) < ε x A hus, for every ε > 0, there exists η > 0 such that, give t, s [0, 1], if t s < η, the xt) xs) wx, η) sup wx, η) < ε x A for every x C. Hece, A is uiformly equicotiuous. Defiitio 2.18 Let, ρ) be a metric space. A set A is relatively compact if its closure, A, is compact. 18

19 Note that a set i A a metric space, ρ) is relatively compact if ad oly if every sequece i that set has a subsequece that coverges i the space the limit is ot ecessarily i the set A itself) 14. heorem 2.19 [2] heorem 7.2) A class A C is relatively compact if ad oly if it satisfies coditios i. ad ii. of Lemma Proof uppose coditios i. ad ii. of Lemma 2.17 hold. he, we kow A is uiformly bouded ad uiformly equicotiuous, which obviously implies that every sequece {x } N A is also uiformly bouded ad uiformly equicotiuous. hus, by the Arzela-Ascoli heorem, every sequece {x } N A has a subsequece that coverges i C, ρ), which implies that A is relatively compact. uppose that A is relatively compact, i.e. the closure A is compact. ice every compact set i a metric space is bouded 15, we kow ) > sup x = sup x A x A sup xt) t [0,1] sup x0), x A which proves i. Cosider ow the sequece of fuctios w, 1 k ) : A R for k N. By Propositio 2.14, every fuctio w, 1 k ) is cotiuous. Moreover, sice every x C is uiformly cotiuous as it is cotiuous o the compact [0, 1] 16 ), by Remark 2.13, for every x A, wx, 1 k ) 0 as k. Lastly, for every i 1 ad x A, we have 1 wx, i+1 ) = sup xt) xs) t s 1 i+1 sup t s 1 i xt) xs) = wx, 1 i ), sice t s < 1 i+i implies that t s < 1 i. Combiig the three properties of {w, 1 k )} metioed i the previous seteces with the fact that A is compact, we ca apply Dii s heorem 17 ad coclude that w, 1 k ) coverges uiformly to 0, i.e. ) lim k his obviously implies that ii. holds. sup x A wx, 1 k ) 0 = 0. heorem 2.20 [2] heorem 7.3) A sequece {P } N of probability measures o C, C) is tight if ad oly if the followig coditios hold. I. For every η > 0, there exists α > 0 ad 1 N such that if 1, the P {x C : x0) α} η. 9) II. For every ε > 0 ad η > 0, there exists δ 0, 1) ad 2 N such that if 2, the P {x C : wx, δ) ε} η. 10) Proof uppose {P } N is tight. hus, for every η > 0, there exists a compact K C such that P K) > 1 η for every N. ice every compact set is closed, ad cosequetly equal to its closure, every compact set is relatively compact. herefore, by heorem 2.19, K satisfies coditios i. ad ii. of Lemma Coditio i. obviously implies that there exists α > 0 such 14 ee [2] heorem p ee [5] Propositio ee [5] heoreme 3.17 Heie heorem) 17 ee [5] heoreme

20 that x0) < α for every x K, i.e. K {x C : x0) < α}. It the follows by mootoicity of probability measures that there exists 1 N, amely 1 = 1, such that if 1, we have P {x C : x0) α} = 1 P {x C : x0) < α} < η, 1 P K) which implies that coditio I. holds. Coditio ii. implies that, for every ε > 0, there exists > 0 such that, for ay 0 < δ, we have sup wx, δ) < ε, x K which obviously implies that wx, δ) < ε for every x K. Hece, for every 0 < δ, K {x C : wx, δ) < ε}. Give that the oly costraits o δ are 0 < δ, we ca choose it such that δ 0, 1) for every > 0. herefore, there exists δ 0, 1) ad 2 N, amely 2 = 1, such that, if 2, which implies that II. holds. P {x C : wx, δ) ε} = 1 P {x C : wx, δ) < ε} < η, 1 P K) Now suppose coditios I. ad II. are satisfied. hus, for every ε > 0 ad η > 0, there exists α > 0, 1 N, δ 0, 1) ad 2 N such that 9) ad 10) hold. ice C, ρ) is separable ad complete, we kow by Remark 1.18 that for every N, P is tight. Hece, {P, P,...} satisfies I. ad II. I other words, for every N, for ay η > 0 ad ε > 0, there exists α > 0 such that ad δ 0, 1) such that P {x C : x0) α } η 11) P {x C : wx, δ ) ε} η. 12) Let 0 = max{ 1, 2 }, α ad δ be defied as i the iequalities 11) ad 12) above, ad α > 0 ad δ > 0 be defied as { } { } α = max α ; max{α } δ = mi δ ; mi {δ } 0 0 Kowig that if a b, the x0) a implies that x0) b, ad if a b, wf, a) ε implies that wf, b) ε, we have the followig iclusios: {x C : x0) α } {x C : x0) α } 0 {x C : x0) α } {x C : x0) α} {x C : wx, δ ) ε} {x C : wx, δ ) ε} 0 {x C : wx, δ ) ε} {x C : wx, δ) ε} hus, applyig mootoicity of probability measures to the iequalities 9),10),11) ad 12) above, we obtai a. For every η > 0, there exists α > 0 such that, for every N, P {x C : x0) α } η; 20

21 b. For every ε > 0 ad η > 0, there exists δ 0, 1) such that for every N P {x C : wx, δ ) ε} η. Let η > 0 be arbitrary. he, there exists α > 0 with the property that the set defied as B := {x C : x0) < α } is such that P B c ) < η for every N. imilarly, we ca fid a sequece {δ k } k 1 0, 1) with the property that the sets B k := {x C : wf, δ k ) < 1 k } are such that P Bk c) < η/2k for k N. Defie K := A, where ) A = B B k. k=1 ice A obviously satisfies coditios i. ad ii. of Lemma 2.17, K is compact. For every N, we have )) P K) P B B k A A for every set A) k=1 )) c ] = 1 P [B B k = 1 P B c 1 > 1 P B c ) + η + his proves that {P } N is tight. k=1 k=1 k=1 B c k )) ) P Bk) c k=1 ) η 2 k = 1 2η Defiitio 2.21 Let Ω, F, P) be a probability space. A radom fuctio o C is a mappig X : Ω C that is F/C-measurable. imilar to the situatio with probability measures ad weak covergece o C, the projectio mappigs ad the set C f of fiite-dimesioal sets determies which mappigs Y : Ω C are radom fuctios ad whe there is weak covergece for distributios of radom fuctios. Propositio 2.22 Let Ω, F, P) be a probability space ad X : Ω C be a arbitrary mappig. he, X is a radom fuctio o C if ad oly if for every k 1 ad U k, the compositio π X : Ω R k is a radom vector 18. Proof If X is a radom fuctio, the every compositio of X with a projectio mappig is a compositio of measurable fuctios, sice it was show i this sectio that every projectio is measurable. It the follows from heorem A.14 that every compositio is a radom vector. uppose ow that every compositio of X with a projectio is a radom vector. Let A C f be a arbitrary fiite-dimesioal set. hus, there exists k 1 ad U k such that A = π 1 H) for some set H B k, which implies that X 1 A) = X 1 π 1 H)) = X 1 π 1 )H) = π X) 1 H) F sice every compositio π X is measurable). As show i the proof of Propositio 2.9, we kow that σc f ) = C, hece, by Propositio A.16, X is F/C-measurable. 18 Recall that for = t 1,..., t k ) U k, π X = Xt 1 ),..., Xt k )) 21

22 heorem 2.23 [2] heorem 7.5) Let X, {X } N be radom fuctios o C defied o a probability space Ω, F, P). If, for every U k ad k 1, d π X π X 13) ad, for every ε > 0, ) lim lim sup PwX, δ) ε) δ 0 = 0, 14) the X d X. Proof Give a arbitrary projectio mappig π, we kow by Defiitio A.42 that the radom vectors π X, {π X } N have respective distributios P π X) 1, {P π X ) 1 } N, which, by associativity of compositios ad compositios of iverse fuctios, are equivalet to P X 1 ) π 1, {P X 1 ) π 1 } N. o improve readability, give U k ad N, we adopt the followig otatio for the remaider of this proof: µ := P X 1 ) π 1, ad µ := P X 1 ) π 1. hus, by the defiitio of covergece i distributio, 13) implies that for every k N ad U k. µ w µ 15) We ow show that the set of probability measures {P X 1 } N satisfies coditios I. ad II. of heorem Restrictig 15) to the case = 0, we have µ w 0 µ 0. Give ay sequece {µ i 0 } i N where the idices i appear i o particular order but are distict, we ca always fid a subsequece {µ i j 0 } j N such that i0 < i1 <..., otherwise the sequece {µ i 0 } i N would obviously ot be ifiite. ice {µ i j 0 } j N is also a subsequece of {µ 0 } N, as the idices are icreasig, it coverges weakly to µ 0. his proves that {µ 0 } N is relatively compact, which implies that it is tight by the coverse part of Prohorov s heorem sice R is separable ad complete). Let η > 0 be arbitrary. he, there exists a compact set K R such that µ 0 K) > 1 η for every N. By the Heie-Borel heorem 19, K is closed ad bouded. hus, there exists α > 0 such that K {x R : x α}. By mootoicity of probability measures, it follows that µ 0 {x R : x α} µ 0 K) for every N. Cosequetly, there exists 1 N, amely 1 = 1, such that, if 1, we have P X 1 {x C : x0) > α} = P π 0 X > α) = µ 0 {x R : x > α} = 1 µ 0 {x R : x α} 1 µ 0 K) < η. his proves that {P X 1 } N satisfies coditio I. of heorem For every ε > 0, 14) implies that, for every η > 0, there exists > 0 such that if 0 < δ, we have ) lim sup PwX, δ) ε) = if sup PwX m, δ) ε) < η. N m his implies that there exists 2 N such that 19 ee [7] Page 105 sup PwX m, δ) ε) = sup P Xm 1 {x C : wx, δ) ε} < η. m 2 m 2 22

23 Hece, if m 2, the P X 1 m ){x C : wx, δ) ε} < η, which proves coditio II. of heorem 2.20, sice δ ca always be chose i 0, 1) ad satisfy 0 < δ. Fially, by heorem 2.20, {P X 1 } N is tight, which, by Corollary 2.11, implies that P X 1 w P X. his proves that X d X. heorem 2.24 [2] Example 5.2) Let {P } N be probability measures o C, C) such that i. For every k 1 ad U k, there exists a probability measure µ o R k, B k ) such that ii. {P } N is relatively compact. P π 1 w µ. he, there exists a probability measure P o C, C) such that P π 1 = µ for every U k, k > 0. Proof ice {P } N is relatively compact, there exists a subsequece {P i } i N that coverges weakly to some probability measure P o C, C). By the cotiuous mappig theorem, for every k N ad U k, we have P i π 1 w P π 1. By i. we have P π 1 w µ, which implies that P i π 1 w µ. hus, by uiqueess of limit for weak covergece, we have P π 1 = µ for every U k, k > 0. Remark 2.25 Notice that, i the previous theorem, it follows by heorem 2.10 that P w P. 2.2 Dosker s heorem Defiitio 2.26 A stochastic process is a collectio of radom variables {X i : i I} defied o the same probability space Ω, F, P). Defiitio 2.27 A stochastic process {W t : t [0, 1]} defied o a give probability space Ω, F, P) is called a Browia motio o [0,1] if it satisfies the followig coditios: i. W starts at zero, i.e. PW 0 = 0) = 1. ii. W has idepedet icremets, i.e. for ay k > 0, for every 0 t 0 <... < t k 1 ad H 0,..., H k B k ) P W ti W ti 1 H i ) = k PW ti W ti 1 H i ). iii. For every 0 s < t 1, W t W s has a ormal distributio with mea 0 ad variace t s, i.e. for every H B, PW t W s H) = 1 2πt s) H e u2 /[2t s)] du. iv. W has cotiuous sample paths, i.e. for ay ω Ω, the fuctio W ω, ) : [0, 1] R defied as W ω, t) = W t ω) for every t [0, 1], is cotiuous everywhere o [0, 1]. 23

24 Propositio 2.28 Let {W t : t [0, 1]} be a Browia motio o a probability space Ω, F, P). For every 0 t 1 <... < t k 1, P W t1,..., W tk ) 1 = P N 1 t1, N 1 t1 + N 2 t2 t 1,..., ) 1 k N i ti t i 1 where t 0 = 0 ad N 1,..., N k are i.i.d. ormally distributed radom variables with mea 0 ad variace 1 defied o a probability space Ω, F, P ). Proof From Defiitio 2.27 ii. ad iii. it is clear that W t1, W t2 W t1..., W tk W tk 1 ) d = N 1 t1, N 2 t2 t 1,..., N k tk t k 1 ). Let g : R s R s be defied as gx 1,..., x s ) = x 1, x 1 + x 2,..., x x s ). he, P W t1,..., W tk ) 1 = P ) 1 W t1, W t2 W t1,..., W tk W tk 1 g 1 [ = P ] 1 gn 1 t1, N 2 t2 t 1,..., N k tk t k 1 ) ) 1 = P k N 1 t1, N 1 t1 + N 2 t2 t 1,..., N i ti t i 1 Defiitio 2.29 A probability measure W o C, C) is called a Wieer measure if it satisfies the followig coditios: i. For every t [0, 1], the radom variable Z t o C, C, W) defied as Z t x) = xt) for every x C is ormally distributed with mea 0 ad variace t, i.e. for every α R. WZ t α) = W{x C : xt) α} =,α] e u2 /2t 2πt ii. he stochastic process {Z t : t [0, 1]} has idepedet icremets, i.e. for every 0 t 0... t k = 1, the radom variables {Z ti Z ti 1 } k are idepedet. he stochastic process {Z t : t [0, 1]} itroduced i the above defiitio is usually referred to as the coordiate-variable process. Defiitio 2.30 Defie the Coordiate-variable radom fuctio o C, C, W), deoted Z : C C, as Zx) = x for every x C. Propositio 2.31 For every 0 s < t 1, Z t Z s has a ormal distributio with mea 0 ad variace t s, i.e. for every H B, 1 PZ t Z s H) = e u2 /[2t s)] du. 2πt s) Proof Let 0 s < t 1 be arbitrary. Lettig ϕ X deote the characteristic fuctio of some radom variable X, it follows from Defiitio 2.29 i. ad Propositio A.59 that ϕ Zs u) = e u2 s 2 ad ϕ Zt u) = e u2 t 2. H du 24

25 Also, ϕ Zt u) = E [ e iuzt] [e iuzt Zs+Zs)] = E ] = E [e iuzt Zs) e iuzs [ = E e iuzt Zs)] E [ e iuzs]. by Defiitio 2.29 ii. ad Propositio A.54) herefore, ϕ Zt Z s = ϕ Z t ϕ Zt = e u2 t s) 2, which implies that the result holds by Propositios A.58 ad A.59. From defiitios 2.27 ad 2.29 aloe, it is ot clear if there exists stochastic processes o [0, 1] that are Browia motios or probability measures o C, C) that are Wieer measures. However, it will oly be ecessary to show the existece of a Wieer measure sice, as we will show, the existece of a Browia motio ca be iferred from the existece of a Wieer measure. Propositio 2.32 If there exists a Wieer measure W, the coordiate-variable process o C, C, W) is a Browia motio. Proof We show that coditios i. ii. iii. ad iv. of Defiitio 2.27 are satisfied. By Defiitio 2.29 i. we kow that Z 0 is ormally distributed with mea 0 ad variace 0. hus, it obviously is the case that WZ 0 = 0) = 1, which proves coditio i. of Defiitio Coditio ii. follows directly from Defiitio 2.29 ii. Coditio iii. is a cosequece of Propositio Coditio iv. is obvious give that sample paths of {Z t : t [0, 1]} are elemets of C ad that every elemet of C is cotiuous by defiitio. Propositio 2.33 If there exists a Wieer measure o C, C), it is uique. Proof Let W ad W be Wieer measures o C, C), Ω, F, P) be a probability space ad {N : Ω R} N be i.i.d. ormally distributed radom variables with mea 0 ad variace 1. Let A C f be arbitrary. By the defiitio of fiite-dimesioal sets, there exists k N, t 1,..., t k ) = U k ad H B k that satisfiy π 1 H) = A. he, by Propositios 2.32 ad 2.28, we have WA) = Wπ 1 H)) = W{x C : xt 1 ),..., xt k )) H} = WZ t1,..., Z tk ) H) = W Z t1,..., Z tk ) 1 H) = P Where t 0 = 0. imilarly, N 1 t1, N 1 t1 + N 2 t2 t 1,..., W A) = W Z t1,..., Z tk ) 1 H) = P N 1 t1, N 1 t1 + N 2 t2 t 1,..., ) 1 k N i ti t i 1 H). ) 1 k N i ti t i 1 H). hus, WA) = W A) for every A C f, which, by Propositio 2.9, implies that W = W. As a cosequece of the above propositio, from ow o, we use the phrase the Wieer measure to emphasize that there caot be more tha oe Wieer measure o C, C). 25

26 Propositio 2.34 he Wieer measure if it exists) is the oly measure W o C, C) such that for ay k N ad t 1,..., t k ) = U k, W π 1 = P N 1 t1, N 1 t1 + N 2 t2 t 1,..., ) 1 k N i ti t i 1 where t 0 = 0 ad N 1,..., N k are i.i.d. ormally distributed radom variables with mea 0 ad variace 1 o a probability space Ω, F, P). Proof Let Q be a probability measure o C, C) such that for every k N ad t 1,..., t k ) = U k, Q π 1 = P = W π 1. N 1 t1, N 1 t1 + N 2 t2 t 1,..., ) 1 k N i ti t i 1 Let A C f. he, there exists l N, U l ad H B l such that π 1 H) = A. herefore, which implies Q = W by Propositio 2.9. QA) = Qπ 1 H)) = Wπ 1 H)) = WA), Defiitio 2.35 Defie the floor fuctio : R Z as x = max{z Z : z x} for every x R ad the ceilig fuctio : R Z as x = mi{z Z : z x} for every x R. Propositio 2.36 For every α R, Proof For every N ad α R, we have α lim = lim α = α. α α = α ad α α = α. For every ε > 0, there exists N N, amely, ay N strictly greater tha 1/ε, such that, if N, the α α α α = 1 1 N < ε. hus, if the limits exist, we have α lim = lim α. Furthermore, for every ε > 0, there exists 1/ε < N N such that if N, the α α = α α sice α ) α α α sice α ) α < ε, which proves the result. 26

27 Defiitio 2.37 Let {ξ } N be a sequece of i.i.d. radom variables with mea 0 ad variace 0 < σ 2 < o a probability space Ω, F, P). Let 0 = 0 ad = ξ ξ for every N. For every t [0, 1] ad iteger 0, defie Xt ω) := 1 σ 1 t ω) + t t ) σ ξ t +1ω). Propositio 2.38 For ay sequece {ξ } N of i.i.d. radom variables with mea 0 ad variace σ 2 < o Ω, F, P) ad every iteger 0, the mappig X : Ω C defied as for every ω Ω is a radom fuctio o C. X ω) : [0, 1] R t X t ω) Proof We first justify the assertio that for every N, X maps Ω ito C. For every ω Ω, X ω) : [0, 1] R is obviously cotiuous, sice it is a liear iterpolatio of the set of poits { )} i, i ω) σ i=0 i the plae [0, 1] R. We ow prove X is F/C-measurable. It was show i the proof of Propositio 2.9 that C = σc f ). herefore, by Propositio A.16, X is F/C-measurable if X ) 1 A) F for every A C f. Let A C f be arbitrary. he, there exists k N, t 1,..., t k ) = U k ad H B k such that π 1 H) = A. We the have X ) 1 A) = X ) 1 π 1 H)) = Xt 1,..., Xt k ) 1 H). Give that, for every i k, Xt i is obviously a radom variable, hece a measurable fuctio, it follows from Propositio A.19 that X ) 1 A) F. Lemma 2.39 [2] heorem 7.4) Let 0 = t 0 <... < t v = 1 be such that for some δ > 0. he, mi t i t i 1 δ 16) 1<i<v I. For every x C, wx, δ) 3 max 1 i v sup xs) xt i 1 ) t i 1 s t i ) II. For every probability measure P o C, C) ad ε > 0, { } v P{x C : wx, δ) 3ε} P x C : sup xs) xt i 1 ) ε t i 1 s t i Proof We first prove I. Let m := max 1 i v sup xs) xt i 1 ) t i 1 s t i ). 27

28 For every x C, if s, t [t i 1, t i ] for some 1 i v, we the have by the triagle iequality that If s [t i 1, t i ] ad t [t i, t i+1 ], the xs) xt) = xs) xt i 1 ) + xt i 1 ) xt) xs) xt i 1 ) + xt) xt i 1 ) 2m. 17) xs) xt) = xs) xt i 1 ) + xt i 1 ) xt i ) + xt i ) xt) xs) xt i 1 ) + xt i ) xt i 1 ) + xt) xt i ) 3m. 18) Furthermore, for every t, s [0, 1] such that t s δ, we kow there exists 1 i < v such that t, s [t i 1, t i ], or s [t i 1, t i ] ad t [t i, t i+1 ], otherwise 16) would be false. Combiig this remark with 17) ad 18) proves coditio I. For II. let ε > 0 ad x {x C : wx, δ) 3ε}. By I. we have that ) m = max 1 i v hus, there exists j v such that { x x C : sup xs) xt i 1 ) t i 1 s t i ε. sup xs) xt j 1 ) ε t j 1 s t j otherwise m would be strictly smaller tha ε. Hece, { } v {x C : wx, δ) 3ε} x C : sup xs) xt i 1 ) ε. t i 1 s t i II. the follows from mootoicity ad coutable subadditivity. }, Lemma 2.40 [2] Lemma p.88) Let {ξ } N be i.i.d. radom variables with mea 0 ad variace 0 < σ 2 < o a probability space Ω, F, P), { } N be defied as i Defiitio 2.37 ad {X } N be defied as i Propositio If lim λ lim sup λ 2 P max i λσ )) i the the set of the distributios {µ X } N is tight. Proof We show that {µ X } N satisfies coditios I. ad II. of heorem For ay ω Ω, we have X ω)0) = X0 ω) = 1 σ 1 0ω) + 0) σ ξ 0+1ω) = 0. = 0, 19) hus, for every α > 0, {ω Ω : X ω)0) α} =. herefore, for every η > 0 ad N, µ X {x C : x0) α} = P ) = 0 < η. his proves {µ X } N satisfies coditio I. of heorem

29 We ow prove that {µ X } satisfies coditio II. of heorem 2.20, i.e. for every ε > 0 ) lim lim sup PwX, δ) ε) = 0. 20) δ 0 For every N ad δ > 0, defie m) := δ ad v) := m) Let N ad δ > 0 be arbitrary. For every iteger i < v) icludig zero), let m i = i m) ad m v) =. We the have 0 = m 0 <... < m v) = ad for every 0 < i < v), m i m i 1 = m), hece, mi m i 1<i<v) m i 1 δ. herefore, it follows from Lemma 2.39 that v) PwX, δ) 3ε) P We otice that, for every iteger t, sup m i 1 k s m i X t = t σ,. X s X m i 1 ) ε. sice t t = 0. Cosequetly, for every s [ m i 1 X m i 1 Xs =, mi mi 1 t σ ], if s = t. for some iteger t, the If there is o such t N, the there exists a iteger t [m i 1, m i ] such that t < s < t +1. ice for every ω Ω, X ω) is liear o every iterval of the form [ t ], we either have X t ω) = t ω) < Xs ω) < t +1ω) = X t +1 ω), X t +1 ω) = t +1ω) < X s ω) < t ω) = X t, t +1 or ω). I either case, it follows that there exists t N either t or t +1, depedig o wether mi 1 ω) is larger or smaller tha Xs ω)) such that X m i 1 ω) Xs ω) mi 1 ω) t ω) σ. Hece, we have that PwX, δ) 3ε) v) P max mi 1 t ε σ ). m i 1 t m i ice the radom variables {ξ } N are i.i.d. we kow by Propositio A.49 that they are statioary. For every l N, defie the mappig Σ : R l R as Σx 1,..., x l ) = x x l. Give i N ad m i 1 t m i, we the have t mi 1 = = t j=1 m i 1 ξ j t j=m i 1+1 j=1 ξ j ξ j = Σξ mi 1+1,..., ξ t ). 29