Converse results for existence of moments and uniform integrability for stopped random walks. (English) Zbl 0607.60055

Let \(S_ n\equiv S(n)=X_ 1+...+X_ n\), \(n\geq 1\), be a random walk on \({\mathbb{R}}\) and N a stopping time for it. Let \(r\geq 1\) and \(E| X_ 1|^ r<\infty\). If either \(EN^ r<\infty\) or \(EX_ 1=0\) and \(EN^ s<\infty\), where \(s=\max (1,r/2)\), then \(E| S_ N|^ r<\infty\) [see the first author, Stopped random walks, limit theorems and applications. (1986), Ch. I]. The paper shows that this still holds for right one-sided moments, if in the second assertion \(EX_ 1=0\) is replaced by \(EX_ 1<0\). The case \(EX_ 1=0\) is still open.
The authors’ main interest is proving converses of these results. Typical examples: Let \(r\geq 1\). If \(E| S_ n|^ r<\infty\), \(E| X_ 1|^ r<\infty\) and \(EX_ 1\neq 0\), then \(EN^ r<\infty\). If \(E| S_ N|^ r<\infty\) and \(EN^ r<\infty\), then \(E| X_ 1|^ r<\infty\). The ”one-sided” problem is completely solved for \(EX_ 1>0\) and \(r>1\) fixed. Out of the 32 combinations of finiteness and infinity of the \(r^{th}\) moments of \(X^+_ 1\), \(X^-_ 1\), N, \(S^+_ N\) and \(S^-_ N\) eighteen are shown to be impossible. For the remaining ones examples are given. Some results are proved for \(0<r<1\), for N not a stopping time and for the special case that \(N,X_ 1,X_ 2,..\). are independent.
Let \(N_ a\), \(a\in A\), be a family of stopping times for the random walk and \(b(a)>0\), \(a\in A\). The paper proves results of the following type. If \(r\geq 1\), \(E| X_ 1|^ r<\infty\), \(EX_ 1=0\) and \(N^ s_ a/b(a)\), \(a\in A\), is uniformly integrable, then so is \(| S(N_ a)|^ r/b(a)\). If \(r\geq 1\), \(E| X_ 1|^ r<\infty\), \(EX_ 1\neq 0\) and \(| S(N_ a)|^ r/b(a)\), \(a\in A\), is uniformly integrable, so is \(N^ r_ a/b(a)\), \(a\in A\).
Reviewer: A.J.Stam


60G50 Sums of independent random variables; random walks
60G40 Stopping times; optimal stopping problems; gambling theory
60F25 \(L^p\)-limit theorems
60K05 Renewal theory
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