  We consider the sums $S_n=\xi_1+\cdots+\xi_n$ of independent identically
distributed random variables. We do not assume that the $\xi$'s have a finite
mean. Under subexponential type conditions on distribution of the summands, we
find the asymptotics of the probability ${\bf P}\{M>x\}$ as $x\to\infty$,
provided that $M=\sup\{S_n,\ n\ge1\}$ is a proper random variable. Special
attention is paid to the case of tails which are regularly varying at infinity.
  We provide some sufficient conditions for the integrated weighted tail
distribution to be subexponential. We supplement these conditions by a number
of examples which cover both the infinite- and the finite-mean cases. In
particular, we show that subexponentiality of distribution $F$ does not imply
subexponentiality of its integrated tail distribution $F^I$.

Probability