Some results associated with random walks - Deligiannidis, Georgios (2010)
In this thesis we treat three problems from the theory and applications of random walks. The first question we tackle is from the theory of the optimal stopping of random walks. We solve the infinite-horizon optimal stopping problem for a class of reward functions admitting a representation introduced in Boyarchenko and Levendorskii [1], and obtain closed expressions for the expected reward and optimal stopping time. Our methodology is a generalization of an early paper by Darling et al. [2] and is based on probabilistic techniques: in particular a path decomposition related to the Wiener-Hopf factorization. Examples from the literature and perturbations are treated to demonstrate the flexibility of our approach. The second question is related to the path structure of lattice random walks. We obtain the exact asymptotics of the variance of the self- intersection local time Vn which counts the number of times the paths of a random walk intersect themselves. Our approach extends and improves upon that of Bolthausen [3], by making use of complex power series. In particular we state and prove a complex Tauberian lemma, which avoids the assumption of monotonicity present in the classical Tauberian theorem. While a bound of order 0(n2) has previously been claimed in the literature ([3], [4]) we argue that existing methods only show the tipper bound O(n2 log n), unless extra conditions are imposed to ensure monotonicity of the underlying sequence. Using the complex Tauberian approach we show that Var (Vn ) Cn2, thus settling a long-standing misunderstanding. Finally, in the last chapter, we prove a functional central limit theorem for one-dimensional random walk in random scenery, a result conjectured in 1979 by Kesten and Spitzer [5]. Essentially random walk in random scenery is the process defined by the partial suins of a collection of random variables (the random scenery), sampled by a random walk. We show that for Z-valued random walk attracted to the symmetric Cauchy law, and centered random scenery with second moments, a functional central limit theorem holds, thus proving the Kesten and Spitzer [5] conjecture which had remained open since 1979. Our proof makes use of tile asymptotic results obtained in the Chapter 3.