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2 Probability and Mathematical Statistics

2 1998

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On the supremum from gaussian processes over infinite horizon

Dębicki Krzysztof, Michna Zbigniew, Rolski Tomasz

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

83--100

In the paper we study the asymptotic of the tail of distribution function P(A(X,c) > x) for x→∞, where A(X, c) is the supremum of X (t)—ct over [0, ∞). In particular, X(t) is the fractional Brownian motion, a nonlinearly scaled Brownian motion or some integrated stationary Gaussian processes. For the fractional Brownian motion we give a stronger result than a recent one of Duffield and O’Connell [5].

Level crossings and local time for regularized gaussian processes

Berzin Corinne, León José R., Ortega Joaquín

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

39--81

Let {X_t, t ∈[0, 1]} be a centred stationary Gaussian process defined on (Q, A, P) with со variance function satisfying r(t) ~ 1 — C|t|^2α, 0<αε = φ_ε * X_ε and Y^ε = X^ε/σ_ε, where σ_ε = varX^ε_t, is a kernel which approaches the Dirac delta function as ε→ 0 and * denotes the convolution. We study the convergence of [formula] where N^v(x) and L_v (x) denote, respectively, the number of crossings and the local time at level x for the process V in [0, 1] and..[formula] The limit depends on the value of α.