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.


60G50 Sums of independent random variables; random walks
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