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lecture-16 - Chapter 16 Convergence of Random Walks This...

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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 semi-group 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 non-overlapping 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) μ t ( w 1 , B ) = B dw 2 1 2 π t e - ( w 2 - w 1 ) 2 2 t (16.1) t ( w 1 , w 2 ) d λ = 1 2 π 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 π t e - ( w 2 - w 1 ) 2 2 t (16.3) We saw in Section 11.1 that the kernels have the semi-group property, so the evolution operators do too. 86
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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 ) 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 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 cadlag sample paths (Theorem 158), but we already knew that, since we know it’s continuous. What we did not know was that the Wiener process is not just Markov but strong Markov, which follows from Theorem 159. It’s easier to find 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 independent N (0 , 1) random variable. (You should convince yourself that this is equivalent.) Now let’s pick an f C 0 which is also twice continuously di ff erentiable, i.e., f C 0 C 2 . Look at K t f ( w ) - f ( w ), and apply Taylor’s theorem, expanding around 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 tZ 2 f (
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