Stability and other limit laws for exit times of random walks from a strip or a halfplane
Consider a random walk \(S_n= X_1+\cdots+ X_n\), \(n\geq 1\), where the \(X_i\), \(i\geq 1\), are i.i.d. real-valued random variables whose common distribution function is not degenerate at \(0\). The authors show that the passage time \(T^*(r)\) of \(S_n\) above a horizontal boundary at \(r\geq 0\) is stable (in probability) in the sense that \(T^*(r)/C(r)@>P>> 1\) as \(r\to\infty\) for a deterministic function \(C(r)> 0\), if and only if the random walk is relatively stable in the sense that \(S_n/B_n @>P>> 1\) as \(n\to\infty\) for a deterministic sequence \(B_n> 0\). (The stability of a passage time is important in some proofs in sequential analysis, where it arises during applications of Anscombe's theorem.) They also prove a counterpart for the almost sure stability of \(T^*(r)\), which is shown to be equivalent to \(E|X|< \infty\), \(EX>0\). Similarly, there are proved counterparts for the exit of the random walk from the strip \(\{|y|\leq r\}\). The conditions are further related to the stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at \(\pm r\) drifts to \(\infty\) as \(r\to\infty\) if and only if the random walk drifts to \(\infty\).
scientific article; zbMATH DE number 1383551