Strong approximation of the St. Petersburg game

Abstract

Let X, X 1 , X 2 , . . . be i.i.d. random variables with P(X = 2 k ) = 2 −k (k ∈ N) and let S n = n k=1 X k . The properties of the sequence S n have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let {Z(t), t ≥ 0} be a semistable Lévy process with underlying Lévy measure k∈Z 2 −k δ 2 k . For a suitable version of (X k ) and Z(t), we prove the strong approximation S n = Z(n) + O(n 5/6+ε ) a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws. ARTICLE HISTORY

Strong approximation and a central limit theorem for St. Petersburg sums

“…is a tail sum of the series representing Y (1) in (1.5) whose tail behavior is described by Theorem 5 of [3]; in particular we have…”

“…It shows also the surprising fact that the partial sum process of (X n ) can be represented as a semistable Lévy process with an asymptotically normal perturbation. In a previous paper [1], a strong approximation of St. Petersburg sums with the weaker remainder term O(n 5/6+ε ) and without the asymptotic normality of the error term was proved by a standard blocking argument. The proof in [1] works for a large class of i.i.d.…”

“…In a previous paper [1], a strong approximation of St. Petersburg sums with the weaker remainder term O(n 5/6+ε ) and without the asymptotic normality of the error term was proved by a standard blocking argument. The proof in [1] works for a large class of i.i.d. sequences (X n ) in the domain of geometric partial attraction of a semistable law G. In contrast, the proof of Theorem 1.1 uses the structure of the St. Petersburg sequences in a substantial way and whether Theorem 1.1 remains valid for a larger class of i.i.d.…”