The rate of escape of random walk
This well worth reading paper is concerned with the rate of escape of a nonnegative random walk \(\{S_ n=X_ 1+...+X_ n\},\) where \(X,X_ 1,X_ 2,..\). is a sequence of i.i.d. nonnegative nondegenerate random variables. The introduction also contains a historical survey in the general case of \({\mathbb{R}}^ d\)-valued random walks. The following two important problems are discussed. First: given a sequence \(\{\beta_ n\}\) of positive real numbers, find a readily explicit sequence \(\{d_ n\}\) depending on \(\{\beta_ n\}\) and the law of X, such that \[ (1)\;P(S_ n\leq \beta_ n\text{ infinitely often) = 0 or 1 according as \(\sum_{n}d_ n\) converges or diverges.} \] The second problem consists of determining whether there is a sequence \(\{\beta_ n\}\) such that \[ (2)\;\liminf_{n\to \infty}S_ n/\beta_ n=1\text{ a.s.} \] The author shows that, in the case when the sums can be normalized so as to be stochastically compact in Feller's sense, and when \(\{n^{-1}\beta_ n\}\) is nondecreasing, then \(d_ n=n^{-1}p_ n \log (3p_ n^{- 1})\), \(p_ n=P(S_ n\leq \beta_ n)\) solves (1). (By a previous result of Jain and Pruitt, the exact asymptotic behavior of \(p_ n\) can be made explicit in terms of the law of X). The author obtains a necessary and sufficient condition in terms of the function \(G(x)=P(X>x)\) for the existence of a sequence \(\{\beta_ n\}\) such that (2) holds. If this condition is not fulfilled, and if \(\{\beta_ n\}\) increases, then \(\liminf_{n\to \infty}S_ n/\beta_ n=0\) or \(\infty\) according as \(\sum_{n}(G(\beta_ n)\vee n^{-1}) \exp -nG(\beta_ n)\) diverges or converges.
rate of escape of a nonnegative random walk
scientific article; zbMATH DE number 4178282