A recent paper by  Pozdnyakov and Steele (2010) is devoted to the so-called binary-plus-passive design. Two problems that the authors do not consider can be identified with the classical gambler’s ruin problem in which delays are allowed.

If Y1,Y2,… is a sequence of random variables such that Yn⟶a.s.Y as n→∞, and {τ(t),t≥0} is a family of “indices” such that τ(t)⟶a.s.∞ as t→∞, then it is pretty obvious that Yτ(t)⟶a.s.Y as t→∞. However, if one relaxes one of ⟶a.s. to ⟶p and lets the other one...

Let Sn, n≥1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller’s famous weak law, Feller (1945), states that Snnlog2n→p1 as n→∞. However, almost sure convergence fails, more precisely, lim supn→∞Snnlog2n=+∞ a.s. and lim...

The laws of large numbers for sums of i.i.d. random variables can be generalized in various ways. The purpose of this note is to collect some domination conditions and to provide a fairly general weak law for arrays. AMS 1980 Subject Classifications: Primary: 60F05, 60F25, 60G42, 60G50

A number of strong limit theorems for renewal counting processes, e.g. the strong law of large numbers, the Marcinkiewicz-Zygmund law of large numbers or the law of the iterated logarithm, can be derived from their corresponding counterparts for the underlying partial sums. In this paper, it is...

The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields. Therein all indices...

The standard assumptions in shock models are that the failure (of a system) is related either to the cumulative effect of a (large) number of shocks or that failure is caused by a shock that exceeds a certain critical level. The present paper is devoted to both types but with variation of the...