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...