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

Allan Gut

Statistics & Probability Letters, 2014, vol. 87, issue C, 105-107

Abstract: 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.

Keywords: Random index; St. Petersburg game; Sums of i.i.d. random variables (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1016/j.spl.2014.01.011

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