- Autor
- TitelStrong approximation of the St. Petersburg game
- Datei
- DOI10.1080/02331888.2016.1269476
- Erschienen inA Journal of Theoretical and Applied Statistics
- Band51
- Erscheinungsjahr2017
- Heft1
- Seiten3-10
- LicenceCC BY
- ISSN1029-4910
- Download Statistik1711
- Peer ReviewJa
- AbstractLet \(X, X_1 , X_2 , . . .\) be i.i.d. random variables with \(P(X = 2^k) = 2^{-k} (k ∈ \mathbb{N}\) 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 \(∑_{k∈\mathbb{Z}} 2^{−k} δ_{2^k}\) . 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.
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