ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: Nagaev, Sergei, Sobolev Institute of Mathematics, Novosibirsk, Russia Probability inequalities for sums of dependent Banach space valued random variables Let X_{k}, k = \overline{1, n}, be random variables taking values in the separable Banach space with the norm |·|. Put S_{k} = ∑_{1}^{k} X_{i}, M_{n} = max_{1≤k≤n} |S_{k}|. Let F_{j}^{k} denote

the σ - lgebra generated by the random variables X_{l}, l = j, ..., k. Define

ϕ^{+}(m) = sup_{1≤k≤n} sup_{A∈F_{1}^{k}, B∈F_{k+m}^{n}}(P(AB)/P(A) − P(B)). Let ϕ^{+}(1) < 1 and

δ > 0 satisfy the condition ρ =: δ + ϕ^{+}(1) < 1. Let α be such that P(2M_{n} > α) < δ. Define Q(r) = ∑_{1}^{n} P(|X_{j}| > r), A_{t} = ∑_{1}^{n} E|X_{j}|^{t}. Theorem. Let r > α and a_{q} ≥ 0 be any sequence such that ∑_{0}^{∞} a_{q} ≤ 1. Then

P(M_{n} > r) < ∑_{q<r/2α} ρ^{q}Q(a_{q}r/2) + (1 − ρ)^{−1}ρ^{r/2α}.

Reference: S. V. Nagaev, Probability inequalities for sums of independent Banach valued random variables, Sibirsk. Mat. Zh. , 28, (1987), 171-184.