Summary
LetT be any set, and let (X t )t ∈ T be a separable gaussian process with mean zero onT. Assume thatX is almost surely bounded, and let\(\mathop {N = \sup }\limits_{ t \in X} {\text{|}}X_t {\text{|}}\).
\(\sigma = \mathop {\sup }\limits_{t \in T} (EX_t^2 )^{1/2}\), and let
Letβ>τ/σ 2. We prove that
Examples are given to show that this result cannot be readily improved. When τ=0, we also show that the distribution ofN has a continuous density with respect to Lebesgue measure.
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Talagrand, M. Sur l'intégrabilité des vecteurs gaussiens. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 1–8 (1984). https://doi.org/10.1007/BF00535169
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DOI: https://doi.org/10.1007/BF00535169