## A necessary condition for making money from fair games.(English)Zbl 0758.60067

There are examples of zero-mean distributions $$F$$ such that if $$X_ 1,X_ 2,\dots$$ are independent random variables with distribution $$F$$, then the random walk $$S_ n=\sum_{j=1}^ n X_ j$$ tends to $$\infty$$ in probability (equivalently $$P\{S_ n>0\}\to 1$$) although it is of course recurrent. If however there is a $$q>1$$ such that $$\int| x|^ q dF(x)<\infty$$, then $$\limsup P\{S_ n>0\}>0$$ and $$\limsup P\{S_ n<0\}>0$$. The main result of the present paper deals with the corresponding question in the case where the $$X_ j$$ are not identically distributed but each $$X_ j$$ has one of two possible non-degenerate, zero-mean distributions $$F_ 1$$, $$F_ 2$$. Suppose there are $$q_ 1$$, $$q_ 2$$ such that $$1<q_ i<2$$ ($$i=1,2$$), $$q_ 1+q_ 2>3$$ and $$\int| x|^{q_ i}dF_ i(x)<\infty$$ ($$i=1,2$$). It is shown that for any sequence $$\sigma(1),\sigma(2),\dots$$ where $$\sigma(j)$$ is either 1 or 2 the following is true: if $$X_ 1,X_ 2,\dots$$ are independent random variables, $$X_ j$$ has distribution $$F_{\sigma(j)}$$ ($$j=1,2,\dots$$) and $$S_ n=\sum_{j=1}^ n X_ j$$, then $$\limsup P\{S_ n>0\}>0$$ and $$\limsup P\{S_ n<0\}>0$$. The proof of the theorem proceeds via a succession of reductions to the case of discrete distributions possessing atoms related in an appropriate manner. The authors also state a number of conjectures connected with the above result.

### MSC:

 60G50 Sums of independent random variables; random walks
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