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Survival probabilities of autoregressive processes

Christoph Baumgarten (2014)

ESAIM: Probability and Statistics

Given an autoregressive process X of order p (i.e. Xn = a1Xn−1 + ··· + apXn−p + Yn where the random variables Y1, Y2,... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to time N (survival or persistence probability). Depending on the coefficients a1,..., ap and the distribution of Y1, we state conditions under which the survival probability decays polynomially, faster than polynomially or converges to a positive constant....

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

Radosław Adamczak, Rafał Latała (2012)

Annales de l'I.H.P. Probabilités et statistiques

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

The fractional mixed fractional brownian motion and fractional brownian sheet

Charles El-Nouty (2007)

ESAIM: Probability and Statistics


We introduce the fractional mixed fractional Brownian motion and fractional Brownian sheet, and investigate the small ball behavior of its sup-norm statistic. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

The magnetization at high temperature for a p-spin interaction model with external field

David Márquez-Carreras (2007)

Applicationes Mathematicae

This paper is devoted to a detailed and rigorous study of the magnetization at high temperature for a p-spin interaction model with external field, generalizing the Sherrington-Kirkpatrick model. In particular, we prove that ⟨ σ i ⟩ (the mean of a spin with respect to the Gibbs measure) converges to an explicitly given random variable, and that ⟨σ₁⟩,...,⟨σₙ⟩ are asymptotically independent.

The majorizing measure approach to sample boundedness

Witold Bednorz (2015)

Colloquium Mathematicae

We describe an alternative approach to sample boundedness and continuity of stochastic processes. We show that the regularity of paths can be understood in terms of the distribution of the argument maximum. For a centered Gaussian process X(t), t ∈ T, we obtain a short proof of the exact lower bound on s u p t ∈ T X ( t ) . Finally we prove the equivalence of the usual majorizing measure functional to its conjugate version.

The Markov property for generalized gaussian random fields

G. Kallianpur, V. Mandrekar (1974)

Annales de l'institut Fourier

We obtain necessary and sufficient conditions in order that a Gaussian process of many parameters (more generally, a generalized Gaussian random field in R n ) possess the Markov property relative to a class of open sets. The method adopted is the Hilbert space approach initiated by Cartier and Pitt. Applications are discussed.

The Rényi distances of Gaussian measures

Jiří Michálek (1999)

Kybernetika

The author in the paper evaluates the Rényi distances between two Gaussian measures using properties of nuclear operators and expresses the formula for the asymptotic rate of the Rényi distances of stationary Gaussian measures by the corresponding spectral density functions in a general case.

Trees and asymptotic expansions for fractional stochastic differential equations

A. Neuenkirch, I. Nourdin, A. Rößler, S. Tindel (2009)

Annales de l'I.H.P. Probabilités et statistiques

In this article, we consider an n-dimensional stochastic differential equation driven by a fractional brownian motion with Hurst parameter H>1/3. We derive an expansion for E[f(Xt)] in terms of t, where X denotes the solution to the SDE and f:ℝn→ℝ is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl.117 (2007) 550–574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift,...

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