Strong approximation of the St. Petersburg game

Let X,X₁,X₂,… be i.i.d. random variables with P(X=2^k)=2^−k (k∈N) and let Sn=∑ⁿ_k=1X_k. 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 ∑_k∈Z2^−kδ₂k. For a suitable version of (X_k) and Z(t), we prove the strong approximation Sn=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.