Strong approximation of the St. Petersburg game

Let X, X1, X2,... be i.i.d. random variables with P(X = 2k)=2-k (k∈ℕ) and let Sn=∑nk=1Xk. The properties of the sequence Sn 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 ∑_kεℤ2^-kδ_2k. 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.