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We investigate compactness properties of weighted summation operators V α,σ as mapping from ℓ 1 (T ) into ℓ q (T ) for some q ∈ (1, ∞). Those operators are defined bywhere T is a tree with induced partial order t s (or s t) for t, s ∈ T . Here α and σ are given weights on T . We introduce a metric d on T such that compactness properties of (T, d) imply two-sided estimates for e n (V α,σ ), the (dyadic) entropy numbers of V α,σ . The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with α(t)σ(t) decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees. 2000

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We study the small deviation problem for a class of symmetric LÃ evy processes, namely, subordinated LÃ evy processes. These processes can be represented as W • A, where W is a standard Brownian motion, and A is a subordinator independent of W . Under some mild general assumption, we give precise estimates (up to a constant multiple in the logarithmic scale) of the small deviation probabilities. These probabilities, also evaluated under the conditional probability given the subordination process A, are formulated in terms of the Laplace exponent of A. The results are furthermore extended to processes subordinated to the fractional Brownian motion of arbitrary Hurst index.

Let X = (X(t)) t∈T be a symmetric α-stable, 0 < α < 2, process with paths in the dual E * of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L α (S, σ) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of φ(ε) = − log P (X E * < ε) as ε → 0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for φ(ε) under an additional assumption on E. As applications we obtain lower small deviation estimates for weighted α-stable Levy motions, linear fractional α-stable motions and d-dimensional α-stable Levy sheets. Our results rest upon an integral representation of L α-valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.