MATH 6605: SUMMARY LECTURE NOTES

MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic process is a collection {X t, t T } of random variables on a probability space Ω, F,. A natural interpretation of the index is time, but many other indices are possible. Example 1. Simple symmetric random walk T = N and oisson rocess T = R +. For each t 1, t 2,..., t k T, the random vector X t1,..., X tk has some distribution. Let s denote this as µ t1,...,t k H = X t1,..., X tk H, H B k, where B k are the Borel sets of R k. The collection of these distributions as the sets t 1, t 2,..., t k T vary {µ t1,...,t k : k N, t 1, t 2,..., t k T, distinct} create the finite dimensional distributions f.d.d. of the process {X t, t T }. Equivalently, we could specify the finite dimensional cumulative distribution functions and still have the same amount of information. For any distinct t 1, t 2,..., t k T, the f.d.d. s satisfy the consistency conditions: C1. µ t1,...,t k H 1... H k = µ ts1,...,t sk H s1... H sk, for H i B, i = 1..., k and any s1,..., sk, a permutation of 1,..., k. C2. µ t1,t 2,...,t k R H 2... H k = µ t2,...,t k H 2... H k for H i B, i = 2..., k. It turns out that any collection of distributions that satisfies the consistency conditions specifies a unique measure for a stochastic process. That is Theorem 1 Kolmogorov Existence Theorem. A family of Borel probability measures {µ t1,...,t k ; k N, t 1,..., t k T distinct} satisfies the consistency conditions C1 and C2 if and only if there exists a unique triple Ω = R T, F T, and random variables {X t ; t T } defined on this triple, such that for all k N, distinct t 1,..., t k T and Borel H R k X t1,..., X tk H = µ t1,...,t k H. Date: March 31, 2008. 1

2 MATH 6605: LECTURE NOTES A key quantity in this result is the σ-algebra, F T. This is the σ-algebra containing the cylinder sets: F T = σ {{X t H}; t T, H R Borel}. One way to think of the cylinder sets is by imagining a finite number of windows at a finite number of times. The cylinder set contains all of the functions which pass through each of the windows. The σ-algebra F T is quite sufficient detailed to answer questions about Ω = R T if T is countable. However, if T is uncountable, R T is much too large and F T is much too small to answer questions of interest. See for example Chapter 36 in [B2], section titled The inadequacy of F T. Example 2. Brownian Motion. Brownian motion has many properties which are quite well-known. One of these, is that it is continuous a.s. However, F T is too small a σ-algebra to ask this question. Instead, we will show that there exists a version of Brownian motion which has continuous sample paths. Let C[0, 1] be the space of continuous functions on the closed interval [0, 1] we could also work with C[0, 10] or C[0, 10 20 ], the choice C[0, 1] is made for convenience. This is a complete separable metric space for the metric dx, y = sup x [0,1] xt yt uniform convergence. The next result tells us what conditions of the finite dimensional distributions guarantee the existence of a process with continuous sample paths. Theorem 2. Let Xt be a stochastic process such that E[ Xt Xs β ] M t s 1+α for some β, α > 0 and M <. Then the stochastic process can be realized on the space of continuous functions, C[0, 1]. The idea of the proof is to show that the polygonal approximation of X n t converges uniformly. The proof appears, for example, in Chapter 5 of [V]. Notably, the rate of the approximation depends only on the parameters M, α, β. Specifically, one can show that there exists a finite constant C = CM, α so that 1.1 sup X n t Xt > γ t [0,1] C γ β 2 nα. Specifically, the theorem tells us that there exists a version of Xt which is continuous a.s. and has the correct finite dimensional distributions. Example 3. Check property for Brownian Motion.

MATH 6605: LECTURE NOTES 3 Therefore we may realize Brownian Motion on the triple Ω = C[0, 1], F T,. Let B denote the σ-algebra generated by the open sets of C[0, 1]. It is not difficult to show that B = F T. This is important in light of how we handle weak convergence see Section 2. Example 4. Define Brownian Bridge. Check for continuity. Theorem 3. Brownian Motion is nowhere differentiable. The proof appears, for example, in Section 7.2 of [D]. 2. Weak Convergence in Metric Spaces This section summarizes the main results on weak convergence in metric spaces. references, see the appropriate sections in [B1, S, V]. For Let X be a separable metric space, and let B denote the σ-algebra generated by the open sets of X. Note that this guarantees that all continuous functions are measurable. A sequence of probability measures on X, B converges weakly, µ n µ, if fdµ n fdµ for all continuous bounded functions f : X R. If Xn is a sequence of random variables on X with LX n = µ n and LX = µ, then the statements X n X and µ n µ are equivalent. Theorem 4. The following are equivalent. 1 µ n µ 2 lim sup n µ n C µc for all closed C 3 lim inf n µ n G µg for all open G 4 lim µ n A = µa for all A B such that µ A = 0. Theorem 5 Continuous Mapping Theorem. Suppose that X n X on X and let g : X R be a functions which is continuous LX a.s.. Then gx n gx. Similar to our definitions before, we say that a sequence {µ n }, of probability measures on X, is tight if for all ε > 0, there exists a compact set K = Kε X such that sup n µ n K c < ε. For X = R d, the definition is the same if we replace the set K with a bounded rectangle. Theorem 6. Suppose that {µ n } is tight and has only one limit point {µ}. Then µ n µ. We have already studied and proved these results for X = R or X = R d. We did not prove the general case in lecture, as they are similar in nature.

4 MATH 6605: LECTURE NOTES 3. Weak Convergence in C[0, 1] Our next goal is to apply the results of Section 2 to weak convergence of stochastic processes in C[0, 1]. By Theorem 6, to check weak convergence of stochastic processes it is enough to check 1 that the sequence of measures is tight, and 2 that the finite dimensional distributions converge. Checking tightness can be quite a work-out at times. We will look at three different ways of doing this. But first, we need to figure out what compact sets in C[0, 1] look like. A family of functions F is equicontinuous if lim sup sup δ 0 x F s t <δ xt xs = 0. This allows us to characterize the compact sets in C[0, 1]. Theorem 7 Ascoli-Arzelá. A set K C[0, 1] has compact closure if it is equicontinuous and sup x K x0 <. 3.1. First Results. An immediate consequence of Theorem 7 is the following. roposition 8. Suppose that X n C[0, 1] and that 1 lim M sup n X n 0 > M = 0 2 For all ɛ > 0, lim δ 0 sup n sup s t δ X n t X n s > ɛ = 0 Then LX n is tight. The result gives a direct brute force approach to checking tightness in many cases. The difficulty of course lies in the presence of the sup inside the probability. The next theorem gives conditions which are much easier to check. Theorem 9. Suppose that µ n = LX n where X n : [0, 1] R. If E[ X n t X n s β ] M t s 1+α for some M < and α, β > 0, all independent of n, and if sup X n 0 > γ 0 n as γ, then {µ n } is a tight sequence of probability measures on C[0, 1]. Sketch of proof. In class I sketched a proof of this result. A key to this was taking as fact 1.1. Let X be any stochastic process which satisfies the conditions of the theorem and let

MATH 6605: LECTURE NOTES 5 X m denote its polygonal approximation on 0, 1/2 m,..., 1. For δ = 2 m Cheb. assumptions sup t s δ X m t X m s > ɛ sup i + 1 i X X > ɛ i 2 m 2 m 2 i + 1 i 2 m sup X X > ɛ i 2 m 2 m 2 [ ] i + 1 i β 2 m sup ɛ β 2 β E X X i 2 m 2 m 2 m ɛ β 2 β M2 m1+α = ɛ β 2 β M2 αm. Therefore, together with 1.1, we obtain that sup t s δ Xt Xs > ɛ sup X m t X m s > ɛ + 2 sup X m t Xs > ɛ t s δ 3 t 3 ɛ β 6 β CM, α3β M + 2 δ α. ɛ β Since the bounds are uniform in any choice of X satisfying the conditions of the theorem, the result follows. Example 5. Check that the continuous polygonal version of the re-scaled random walk 1 n X nt converges to Brownian motion. 3.2. Empirical rocesses. Let X 1, X 2,... be iid random variables with some cumulative distribution function F. The empirical distribution function is defined as F n X = 1 n IX i x. n i=1 This quantity is of key interest in statistics. Our goal is to show the following two results. Theorem 10 Glivenko-Cantelli. sup F n x F x 0, a.s. x

6 MATH 6605: LECTURE NOTES Theorem 11 Donsker. Let Ux denote the Brownian Bridge. Then sup nf n x F x sup UF x. x x One can also show that sup Ut α = 1 2 1 k+1 e 2k2 α 2. 0t1 k=1 In this section we will prove the above results by showing Theorem 12. Let Ux denote the Brownian Bridge, and suppose that F x = x. That is, we are sampling from the Uniform[0,1] distribution. Then Y n x { nf n x x} {UF x}. Exercise 1. Show that Theorem 13 implies Theorems 10 and 11. Exercise 2. Show that Theorem 13 holds in finite dimensional distributions. As before, to prove Theorem 13, we need to first place F n in the space of continuous functions. To do this, define G n to be the distribution function corresponding to a uniform distribution of mass n + 1 1 over the intervals [X i, X i+1 ] for i = 1,..., n, where X 0 = 0 and X n+1 = 1. It is not difficult to show that 3.2 F n x G n x 1 n, 0 x 1. The theorem we will prove is the following. Theorem 13. Let Ux denote the Brownian Bridge, and suppose that F x = x. That is, we are sampling from the Uniform[0,1] distribution. Then Z n x { ng n x x} {Ux}. Since we already have convergence of finite dimensional distributions exercise, by 3.2 and Theorem 8 [correction : actually, I should refer to Theorem 16 here], to prove tightness it remains to show that for all η > 0 sup Y n t Y n s > η t s δ as δ 0. Because F x = x, it is enough to check sup Y n t > η t δ 0. 0

Define the quantities Since, M n M n + Y n δ, MATH 6605: LECTURE NOTES 7 M m = max nδi/m, Y n δ Y n δi/m } 1im M m = max nδi/m }. 1im M m > η M m > η/2 + Y n δ > η/2. We will show that M m > η/2 Bη 4 δ 2 in a little bit. By right-continuity of F n and Chebyshev, we can let m in the above to get that sup Y n t > η t δ Bη 4 δ 2 + 24 η 4 E[ Y nδ 4 ]. Noting that lim n E[ Y n δ 4 ] 3δ 2 completes the proof. Theorem 14. Let ξ 1, ξ 2,... be random variables not necessarily iid, and define S k = ξ 1 + ξ 2 +... + ξ k, with S 0 = 0. Let M m = max 0km min { S k, S m S k }. Suppose there exists non-negative numbers u 1, u 2,... such that E [ 2 3.3 S j S i 2 S k S j 2] u l. Then there exists a constant K 1, large enough so that 22 2/5 + K 1/5 5 1, satisfying i<lk M m λ K λ 4 u 1 +... u m 2 Exercise 3. Show that Theorem 14 implies the remaining bound M m > η/2 Bη 4 δ 2, for some positive B <. That is, let ξ i = Y n δi/m Y n δi 1/m, and show that in this case S k satisfies condition 3.3. 3.3. Using Martingale Theory. I will omit the write up of the basic martingale stuff, since this is not only fairly straightforward, but also appears in Chapter 14 of [R]. The key result that we are interested in is Theorem 15. If {X n } is a submartingale, then for all α > 0 [ ] max X i α E[ X n ]. 0in α

8 MATH 6605: LECTURE NOTES In this section we will apply this result to provide an alternative proof of Example 5. First though, I need to address a technical issue. I realize that roposition 8 is a little too stringent for our purposes, and so we will next state a slightly easier version. roposition 16. Suppose that X n C[0, 1] and let µ n = LX n. The sequence {µ n } is tight if and only if 1 lim M sup n X n 0 > M = 0 2 For all ɛ, η > 0, there exists a δ 0, 1, and an integer n 0 such that sup n n 0 sup X n t X n s > ɛ s t δ Sketch of proof. The result follows from roposition 8 and the fact that any random variable X C[0, 1] has a tight measure. This last fact is a bit technical, and its proof appears, for example, in [B1], page 10, Theorem 1.4. Therefore, by the Arzela-Ascoli result, for any fixed n, lim δ 0 sup X n t X n s > ɛ s t δ Now, since each µ 1, µ 2,..., µ n0 is tight, we can strengthen condition 2 above by decreasing δ is necessary, to obtain condition 2 of roposition 8. Example 6. Let X n be a symmetric random walk. Using roposition 16 and Theorem 15, show that 1 n X nt converges to Brownian motion. η. = 0. 4. Continuous-time Martingales Consider a stochastic process {X t } t 0, defined on Ω, F,. A filtration is a family of σ algebras, {F t } t 0, such that F s F t F, for all s t. The process X t is a martingale with respect to the filtration {F t }, if 1 E[ X t ] < 2 E[X t F s ] = X s for all s t. As before, if the equality is replaced with then we have a submartingale. If it is replaced with a then we have a supermartingale. Example 7. B t and B 2 t t are both martingales with respect to the canonical filtration of Brownian motion, F t = σx u, u t. Example 8. A stochastic process, X t, is a Brownian motion if and only if for all λ, Z t = exp{λx t λ 2 t/2} is a martingale.

MATH 6605: LECTURE NOTES 9 Exercise 4. Let N t be a oisson process with rate λ. Find function at, bt that N t at and N 2 t bt are both martingales with F t = σn u, u t. A non-negative random variable is a stopping time with respect to a filtration, {F t } t 0, if for each t 0, {ω, : τω t} F t. That is, by time t, you know whether or not you have stopped. There is also a continuous version of the optional sampling theorem. Theorem 17 Optional Sampling Theorem. Let M t be a right-continuous martingale with respect to {F t } t 0, and let τ, σ be two stopping times such that 0 σ τ M <. Then E[M σ ] = E[M τ ]. In particular, it follows that if τ is a bounded stopping time, then E[M τ ] = E[M 0 ]. Note that the theorem does not hold if the stopping time is not bounded. This idea may be used to find the distribution of the hitting time of Brownian motion. Exercise 5. Let τ = inf{t 0 : B t 1}. Then τ is not a bounded stopping time, but τ n is, for any n. Therefore, we may apply the optional sampling theorem to the martingale in Example 8 and τ n. Letting n go to infinity using the BCT, we obtain that E [ exp{ λ 2 τ/2} ] = exp{ λ}. A transformation and inversion of the Laplace transform gets us to τ t = t 0 e 1 2u 2πu 3 du. Another way of computing this distribution is through the reflection principle. Theorem 18. If B t is a Brownian motion starting at 0, then for any a > 0 sup B s a 0st = 2 B t a. Here, we have simply that τ t = sup 0st B s 1 = 2 B t 1, and a simple transformation yields the result. Another application albeit much more tedious of the reflection principle is the proof of the next result. Theorem 19. Let Ut denote a Brownian Bridge. Then sup Us b = 1 2 0s1 k 1 1 k+1 e 2k2 b 2. The proof appears in, for example, [D].

10 MATH 6605: LECTURE NOTES References [B1] atrick Billingsley; Convergence of robability Measures, Wiley, 1968. [B2] atrick Billingsley; robability and Measure, 3rd Edition, Wiley, 1995. [D] Rick Durrett; robability: Theory and Examples, 3rd Edition, Duxbury ress, 2005. [R] Jeff Rosenthal; A First Look at Rigorous robability Theory. 2nd Edition, World Scientific, 2006. [S] Galen Shorack; robability for Statisticians, Springer, 2000. [V] S.R.S. Varadhan; Stochastic rocesses, Courant, 1968. repared by Hanna K. Jankowski Department of Mathematics and Statistics, York University e-mail: hkj@mathstat.yorku.ca