Convergence of Random Walks

Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of the theory of Feller processes. Section 16.1 fins the semi-group of the Wiener process, shows it satisfies the Feller properties, an fins its generator. Section 16.2 turns ranom walks into calag processes, an gives a fairly easy proof that they converge on the Wiener process. 16.1 The Wiener Process is Feller Recall that the Wiener process W (t) is efine by starting at the origin, by inepenent increments over non-overlapping intervals, by the Gaussian istribution of increments, an by continuity of sample paths (Examples 38 an 78). The process is homogeneous, an the transition kernels are (Section 11.1) µ t (w 1, B) = µ t (w 1, w 2 ) λ = B w 2 1 2πt e (w 2 w 1 )2 2t (16.1) 1 2πt e (w 2 w 1 )2 2t (16.2) where the secon line gives the ensity of the transition kernel with respect to Lebesgue measure. Since the kernels are known, we can write own the corresponing evolution operators: 1 K t f(w 1 ) = w 2 f(w 2 ) e (w 2 w 1 )2 2t (16.3) 2πt We saw in Section 11.1 that the kernels have the semi-group property, so the evolution operators o too. 86

CHAPTER 16. CONVERGENCE OF RANDOM WALKS 87 Let s check that {K t }, t 0 is a Feller semi-group. The first Feller property is easier to check in its probabilistic form, that, for all t, y x implies W y (t) W x (t). The istribution of W x (t) is just N (x, t), an it is inee true that y x implies N (y, t) N (x, t). The secon Feller property can be checke in its semi-group form: as t 0, µ t (w 1, B) approaches δ(w w 1 ), so lim t 0 K t f(x) = f(x). Thus, the Wiener process is a Feller process. This implies that it has calag sample paths (Theorem 158), but we alreay knew that, since we know it s continuous. What we i not know was that the Wiener process is not just Markov but strong Markov, which follows from Theorem 159. It s easier to fin the generator of {K t }, t 0, it will help to re-write it in an equivalent form, as K t f(w) = E f(w + Z ] t) (16.4) where Z is an inepenent N (0, 1) ranom variable. (You shoul convince yourself that this is equivalent.) Now let s pick an f C 0 which is also twice continuously ifferentiable, i.e., f C 0 C 2. Look at K t f(w) f(w), an apply Taylor s theorem, expaning aroun w: K t f(w) f(w) = E f(w + Z ] t) f(w) (16.5) = E f(w + Z ] t) f(w) (16.6) = E Z tf (w) + 1 2 tz2 f (w) + R(Z ] t) (16.7) K t f(w) f(w) lim t 0 t = tf (w)e Z] + t f (w) 2 = 1 ] 2 f E R(Z t) (w) + lim t 0 t E Z 2] + E R(Z ] t) (16.8) (16.9) So, we nee to investigate the behavior of the remainer term R(Z t). We know from Taylor s theorem that R(Z t) = tz2 2 1 0 u f (w + uz t) f (w) (16.10) (16.11) Since f C 0 C 2, we know that f C 0. Therefore, f is uniformly continuous, an has a moulus of continuity, m(f, h) = sup f (x) f (y) (16.12) x,y: x y h which goes to 0 as h 0. Thus R(Z t) tz2 2 m(f, Z t) (16.13) R(Z t) Z 2 m(f, Z t) lim lim (16.14) t 0 t t 0 2 = 0 (16.15)

CHAPTER 16. CONVERGENCE OF RANDOM WALKS 88 Plugging back in to Equation 16.9, Gf(w) = 1 ] 2 f E R(Z t) (w) + lim t 0 t 2 (16.16) = 1 2 f (w) (16.17) That is, G = 1 2 w, one half of the Laplacian. We have shown this only for 2 C 0 C 2, but this is clearly a linear subspace of C 0, an, since C 2 is ense in C, it is ense in C 0, i.e., this is a core for the generator. Hence the generator is really the extension of 1 2 2 w to the whole of C 2 0, but this is too cumbersome to repeat all the time, so we just say it s the Laplacian. 16.2 Convergence of Ranom Walks Let X 1, X 2,... be a sequence of IID variables with mean 0 an variance 1. The ranom walk process S n is then just n i=1 X i. It is a iscrete-time Markov process, an consequently also a strong Markov process. Imagine each step of the walk takes some time h, an imagine this time interval becoming smaller an smaller. Then, between any two times t 1 an t 2, the number of steps of the ranom walk will be about t2 t1 h, which will go to infinity. The isplacement of the ranom walk between t 1 an t 2 will then be a sum of an increasingly large number of IID ranom variables, an by the central limit theorem will approach a Gaussian istribution. Moreover, if we look at the interval of time from t 2 to t 3, we will see another Gaussian, but all of the ranom-walk steps going into it will be inepenent of those going into our first interval. So, we expect that the ranom walk will in some sense come to look like the Wiener process, no matter what the exact istribution of the X 1. Let s consier this in more etail. Define Y n (t) = n 1/2 nt] i=0 X i = n 1/2 S nt], where X 0 = 0 an nt] is the integer part of the real number nt. You shoul convince yourself that this is a Markov process, with calag sample paths. We want to consier the limiting istribution of Y n as n. First of all, we shoul convince ourselves that a limit istribution exists. But this is not too har. For any fixe t, Y n (t) approaches a Gaussian istribution by the central limit theorem. For any fixe finite collection of times t 1 t 2... t k, Y n (t 1 ), Y n (t 2 ),... Y n (t k ) approaches a limiting istribution if Y n (t 1 ), Y n (t 2 ) Y n (t 1 ),... Y n (t k ) Y n (t k 1 ) oes, but that again will be true by the (multivariate) central limit theorem. Since the limiting finite-imensional istributions exist, some limiting istribution exists (via Theorem 23). It remains to ientify it. Lemma 173 Y n f W.

CHAPTER 16. CONVERGENCE OF RANDOM WALKS 89 Proof: For all n, Y n (0) = 0 = W (0). For any t 2 > t 1, L (Y n (t 2 ) Y n (t 1 )) = L 1 nt 2] n i=nt 1] X i (16.18) N (0, t 2 t 1 ) (16.19) = L (W (t 2 ) W (t 1 )) (16.20) Finally, for any three times t 1 < t 2 < t 3, Y n (t 3 ) Y n (t 2 ) an Y n (t 2 ) Y n (t 1 ) are inepenent for sufficiently large n, because they become sums of isjoint collections of inepenent ranom variables. Thus, the limiting istribution of Y n starts at the origin an has inepenent Gaussian increments. Since these properties etermine the finite-imensional istributions of the Wiener process, f Y n W. Theorem 174 Y n W. f Proof: By Theorem 165, it is enough to show that Y n W, an that any of the properties in Theorem 166 hol. The lemma took care of the finite-imensional convergence, so we can turn to the secon part. A sufficient conition is property (1) inn the latter theorem, that Y n (τ n +h n ) Y n (τ n ) P 0 for all finite optional times τ n an any sequence of positive constants h n 0. Y n (τ n + h n ) Y n (τ n ) = n 1/2 Snτn+nh n] S nτn] (16.21) = n 1/2 Snhn] S 0 (16.22) = n 1/2 Snhn] (16.23) nh n] = n 1/2 X i (16.24) To see that this converges in probability to zero, we will appeal to Chebyshev s inequality: if Z i have common mean 0 an variance σ 2, then, for every positive ɛ, ( ) m P Z i > ɛ mσ2 ɛ 2 (16.25) i=1 Here we have Z i = X i / n, so σ 2 = 1/n, an m = nh n ]. Thus ( P n ) 1/2 S nhn] > ɛ i=0 nh n] nɛ 2 (16.26) As 0 nh n ]/n h n, an h n 0, the bouning probability must go to zero for every fixe ɛ. Hence n 1/2 S nhn] P 0.

CHAPTER 16. CONVERGENCE OF RANDOM WALKS 90 Corollary 175 (The Invariance Principle) Let X 1, X 2,... be IID ranom variables with mean µ an variance σ 2. Then Y n (t) 1 nt] n i=0 X i µ σ W (t) (16.27) Proof: (X i µ)/σ has mean 0 an variance 1, so Theorem 174 applies. This result is calle the invariance principle, because it says that the limiting istribution of the sequences of sums epens only on the mean an variance of the iniviual terms, an is consequently invariant uner changes which leave those alone. Both this result an the previous one are known as the functional central limit theorem, because convergence in istribution is the same as convergence of all boune continuous functionals of the sample path. Another name is Donsker s Theorem, which is sometimes associate however with the following corollary of Theorem 174. Corollary 176 (Donsker s Theorem) Let Y n (t) an W (t) be as before, but restrict the inex set T to the unit interval 0, 1]. Let f be any function from D(0, 1]) to R which is measurable an a.s. continuous at W. Then f(y n ) f(w ). Proof: Exercise. This version is especially important for statistical purposes, as we ll see a bit later. 16.3 Exercises Exercise 16.1 Go through all the etails of Example 138. a Show that F X t F W t for all t, an that F X F W. b Show that τ = inf t X(t) = (0, 0) is a F X -optional time, an that it is finite with probability 1. c Show that X is Markov with respect to both its natural filtration an the natural filtration of the riving Wiener process. Show that X is not strongly Markov at τ. e Which, if any, of the Feller properties oes X have? Exercise 16.2 Consier a -imensional Wiener process, i.e., an R -value process where each coorinate is an inepenent Wiener process. Fin the generator. Exercise 16.3 Prove Donsker s Theorem (Corollary 176).

CHAPTER 16. CONVERGENCE OF RANDOM WALKS 91 Exercise 16.4 (Diffusion equation) As mentione in class, the partial ifferential equation 1 2 f(x, t) f(x, t) 2 x 2 = t is calle the iffusion equation. From our iscussion of initial value problems in Chapter 12 (Corollary 126 an relate material), it is clear that the function f(x, t) solves the iffusion equation with initial conition f(x, 0) if an only if f(x, t) = K t f(x, 0), where K t is the evolution operator of the Wiener process. x2 a Take f(x, 0) = (2π10 4 ) 1/2 e 2 10 4. f(x, t) can be foun analytically; o so. b Estimate f(x, 10) over the interval 5, 5] stochastically. Use the fact that K t f(x) = E f(w (t)) W (0) = x], an that ranom walks converge on the Wiener process. (Be careful that you scale your ranom walks the right way!) Give an inication of the error in this estimate. c Can you fin an analytical form for f(x, t) if f(x, 0) = 1 0.5,0.5] (x)? Fin f(x, 10), with the new initial conitions, by numerical integration on the omain 10, 10], an compare it to a stochastic estimate.