Summary
Let (Ω,ℱ,P) be a probability space and let {itX n (ω)} n=1 be a sequence of i.i.d. random vectors whose state space isZ m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS n (ω) be the random walk defined by\(S_n (\omega ) = \sum\limits_{j = 1}^n {X_j (\omega )}\). ForxεZ m andα∈U m, them-dimensional torus, let\(\left\langle {\alpha ,x} \right\rangle = e^{2\pi i} \sum\limits_{j = 1}^m {\alpha _j x_j }\). Finally let\(\phi (\alpha ) = E\{ \left\langle {\alpha ,X_1 (\omega )} \right\rangle \}\) be the characteristic function of the X's.
In this paper we show that, under mild restrictions, there exists a setΩ ⊂Ω withP{Ω 0 } = 1 such that forω ∈ Ω 0 we have\(\mathop {\lim }\limits_{n \to \infty } \left| {\frac{1}{n}\sum\limits_{j = 1}^n {\left\langle {\alpha ,S_j (\omega )} \right\rangle } } \right| = 0\) for all aα∈U m,αle0.
As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.
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Research supported by N.S.F. Grant # MCS 77-26809
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Blum, J.R., Reich, J.I. p-Sets for random walks. Z. Wahrscheinlichkeitstheorie verw Gebiete 48, 193–200 (1979). https://doi.org/10.1007/BF01886872
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DOI: https://doi.org/10.1007/BF01886872