On convergence of randomly indexed sequences; a counterexample based on the St. Petersburg game

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 remain as is, then one of the resulting conclusions holds, whereas the other one does not. In this note we provide a “more natural” counterexample than the original one due to Richter (1965), followed by a minor extension.