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...