Abstract
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-kδ2k. For a suitable version of (Xk) and Z(t), we prove the strong approximation Sn=Z(n)+O(n5/6+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.
Original language | English |
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Pages (from-to) | 3-10 |
Number of pages | 8 |
Journal | Statistics |
Volume | 51 |
Issue number | 1 |
DOIs | |
Publication status | Published - 3 Jan 2017 |
Keywords
- a.s. invariance principle
- semistable laws
- St. Petersburg game
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty