## 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

### MSC:

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

### Keywords:

uniform integrability; stopping times
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