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On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model

Franco Flandoli, Massimiliano Gubinelli, Francesco Russo (2009)

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

We study the pathwise regularity of the map φ↦I(φ)=∫0T〈φ(Xt), dXt〉, where φ is a vector function on ℝd belonging to some Banach space V, X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture...

On the strong Brillinger-mixing property of α -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function C ( x , y ) defining an α -determinantal point process (DPP). Assuming absolute integrability of the function C 0 ( x ) = C ( o , x ) , we show that a stationary α -DPP with kernel function C 0 ( x ) is “strongly” Brillinger-mixing, implying, among others, that its tail- σ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications...

On the Structure of Spatial Branching Processes

Matthes, Klaus, Nawrotzki, Kurt, Siegmund-Schultze, Rainer (1997)

Serdica Mathematical Journal

The paper is a contribution to the theory of branching processes with discrete time and a general phase space in the sense of [2]. We characterize the class of regular, i.e. in a sense sufficiently random, branching processes (Φk) k∈Z by almost sure properties of their realizations without making any assumptions about stationarity or existence of moments. This enables us to classify the clans of (Φk) into the regular part and the completely non-regular part. It turns out that the completely non-regular branching...

On the supremum of random Dirichlet polynomials

Mikhail Lifshits, Michel Weber (2007)

Studia Mathematica

We study the supremum of some random Dirichlet polynomials D N ( t ) = ∑ n = 2 N ε ₙ d ₙ n - σ - i t , where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials ∑ n ∈ τ ε ₙ n - σ - i t , τ = 2 ≤ n ≤ N : P ⁺ ( n ) ≤ p τ , P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, s u p t ∈ ℝ | ∑ n = 2 N ε ₙ n - σ - i t | ≈ ( N 1 - σ ) / ( l o g N ) . The proofs are entirely based on methods of stochastic processes, in particular the metric...

On the tail index estimation of an autoregressive Pareto process

Marta Ferreira (2013)

Discussiones Mathematicae Probability and Statistics

In this paper we consider an autoregressive Pareto process which can be used as an alternative to heavy tailed MARMA. We focus on the tail behavior and prove that the tail empirical quantile function can be approximated by a Gaussian process. This result allows to derive a class of consistent and asymptotically normal estimators for the shape parameter. We will see through simulation that the usual estimation procedure based on an i.i.d. setting may fall short of the desired precision.

On the tails of the distribution of the maximum of a smooth stationary gaussian process

Jean-Marc Azaïs, Jean-Marc Bardet, Mario Wschebor (2002)

ESAIM: Probability and Statistics

We study the tails of the distribution of the maximum of a stationary gaussian process on a bounded interval of the real line. Under regularity conditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [11] for a sufficiently small interval.

Currently displaying 1941 – 1960 of 3391