Vladimir I. Piterbarg

Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = supσ≥t(X(σ) - X(t) - c(σ - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.

On the supremum of γ-reflected processes with fractional Brownian motion as input

Abstract Let { X H ( t ) , t ≥ 0 } be a fractional Brownian motion with Hurst index H ∈ ( 0 , 1 ] and define a γ -reflected process W γ ( t ) = X H ( t ) − c t − γ inf s ∈ [ 0 , t ] ( X H ( s ) − c s ) , t ≥ 0 with c > 0 , γ ∈ [ 0 , 1 ] two given constants. In this paper we establish the exact tail asymptotic behaviour of M γ ( T ) = sup t ∈ [ 0 , T ] W γ ( t ) for any T ∈ ( 0 , ∞ ] . Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

High excursions for nonstationary generalized chi-square processes

Suppose that X(t), t[set membership, variant][0, T], is a centered differentiable Gaussian random process, X1(t), ..., Xn(t) are independent copies of X(t). An exact asymptotic behavior of large deviation probabilities for the process , where b1, b2, ... bn are positive constants is investigated. It is assumed that the variance of the process attains its global maximum in only one inner point of the interval [0, T], with a nondegeneracy condition.

Spatial-correlation-coefficient at the basestation, in closed-form explicit analytic expression, due to heterogeneously Poisson distributed scatterers

This letter analytically derives, for stochastically located scatterers, closed-form expressions of the uplink received-signals spatial-correlation-coefficient across the basestation antenna arrays spatial aperture, explicitly as a function of 1) the basestations interantenna spacing, and 2) each scatterer-clusters distance and spatial angle from the basestation. This rigorous analytical derivation is based on idealized geometric relationships among the mobile transmitter, the basestations receiving antennas, and the scatterers (whose spatial distribution is modeled as Poisson-distributed with a multi-Gaussian heterogeneous intensity).

On Maximum of Gaussian Non-Centered Fields Indexed on Smooth Manifolds

The double sum method of evaluation of probabilities of large deviations for Gaussian processes with non-zero expectations is developed. Asymptotic behaviors of the tail of non-centered locally stationary Gaussian fields indexed on smooth manifolds are evaluated. In Particular, smooth Gaussian fields on smooth manifolds are considered.

Mathematical Methods and Concepts for the Analysis of Extreme Events

Mathematical tools for the analysis of Xevents, maxima of processes and rare events are presented. Methods and concepts of classical statistical extreme value theory are described, as well as those of large deviation theory. Techniques from other areas such as statistical mechanics, the theory of dynamical systems and the theory of singularities are also briefly discussed.

On the asymptotic Laplace method and its application to random chaos

The asymptotics of the multidimensional Laplace integral for the case in which the phase attains its minimum on an arbitrary smooth manifold is studied. Applications to the study of the asymptotics of the distribution of Gaussian and Weibullian random chaoses are considered.

Asymptotic Behavior of Reliability Function for Multidimensional Aggregated Weibull Type Reliability Indices

We derive asymptotic approximation of high risk probability (ruin probability) for multidimensional aggregated reliability index which is a linear combination of single independent indexes, whose reliability functions (distribution tails) behave like Weibull tails.

Large extremes of Gaussian chaos processes

We study probabilities of large extremes of Gaussian chaos processes, that is, homogeneous functions of Gaussian vector processes. Important examples are products of Gaussian processes and quadratic forms of them. Exact asymptotic behaviors of the probabilities are found. To this aim, we use joint results of E. Hashorva, D. Korshunov and the author on Gaussian chaos, as well as a substantially modified asymptotical Double Sum Method.

Asymptotic expansions in the Poisson limit theorem for large excursions of stationary Gaussian sequences

There exist two approaches to the proof of the Poisson limit theorem for the events {X (n)~ u}, u-+ oo, where X(n) is a stationary Gaussian sequence: the method of moments [i] and the method of comparison [2]. The latter method makes it possible to estimate the rate of convergence and to find the correction terms. We realize this possibility in the present note with the aid of a further advance in the method of comparison proposed in [3].