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Unformatted text preview: Chapter 16 Convergence of Random Walks This lecture examines the convergence of random walks to the Wiener process. This is very important both physically and statis tically, and illustrates the utility of the theory of Feller processes. Section 16.1 finds the semigroup of the Wiener process, shows it satisfies the Feller properties, and finds its generator. Section 16.2 turns random walks into cadlag processes, and 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 defined by starting at the origin, by independent increments over nonoverlapping intervals, by the Gaussian distri bution of increments, and by continuity of sample paths (Examples 38 and 78). The process is homogeneous, and the transition kernels are (Section 11.1) t ( w 1 , B ) = B dw 2 1 2 t e ( w 2 w 1 ) 2 2 t (16.1) d t ( w 1 , w 2 ) d = 1 2 t e ( w 2 w 1 ) 2 2 t (16.2) where the second line gives the density of the transition kernel with respect to Lebesgue measure. Since the kernels are known, we can write down the corresponding evolution operators: K t f ( w 1 ) = dw 2 f ( w 2 ) 1 2 t e ( w 2 w 1 ) 2 2 t (16.3) We saw in Section 11.1 that the kernels have the semigroup property, so the evolution operators do too. 86 CHAPTER 16. CONVERGENCE OF RANDOM WALKS 87 Lets check that { K t } , t 0 is a Feller semigroup. The first Feller property is easier to check in its probabilistic form, that, for all t , y x implies W y ( t ) d W x ( t ). The distribution of W x ( t ) is just N ( x, t ), and it is indeed true that y x implies N ( y, t ) N ( x, t ). The second Feller property can be checked in its semigroup form: as t 0, t ( w 1 , B ) approaches ( w w 1 ), so lim t K t f ( x ) = f ( x ). Thus, the Wiener process is a Feller process. This implies that it has cadlag sample paths (Theorem 158), but we already knew that, since we know its continuous. What we did not know was that the Wiener process is not just Markov but strong Markov, which follows from Theorem 159. Its easier to find the generator of { K t } , t 0, it will help to rewrite it in an equivalent form, as K t f ( w ) = E f ( w + Z t ) (16.4) where Z is an independent N (0 , 1) random variable. (You should convince yourself that this is equivalent.) Now lets pick an f C which is also twice continuously differentiable, i.e., f C C 2 . Look at K t f ( w ) f ( w ), and apply Taylors theorem, expanding around w :...
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This note was uploaded on 12/20/2011 for the course STAT 36754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
 Spring '06
 Schalizi

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