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Some limit theorems on the increments of a multi-parameter fractional Brownian motion. (English) Zbl 0990.60025
Almost sure upper and lower bounds are proved for the maximum increments (over rectangles) of a \(d\)-parameter fractional Brownian motion, when the size of the rectangles satisfies some mild conditions. The normalization is sharp, i.e. the bounds coincide under the joint assumptions. The proofs are based on a suitable large deviation upper bound for the maximum increments above which is first derived from a version of X. Fernique’s lemma [C. R. Acad. Sci., Paris 258, 6058-6060 (1964; Zbl 0129.30101)].

MSC:
60F15 Strong limit theorems
60G15 Gaussian processes
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