The hierarchy of chaotic properties of symmetric infinitely divisible stationary processes is studied in the language of their stochastic representation. The structure of the Musielak-Orlicz space in this representation is exploited here.

For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = μ∘λ: λ ∈ , which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for which ℳ(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ₁, Θ₂ independent of X, X’ there exists a random variable Θ independent of X such...

We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the gaussian and the purely non-gaussian parts of the infinitely divisible limit. We also discuss...

We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the Gaussian and the purely non-Gaussian parts of the infinitely divisible limit. We also discuss...

We consider the random recursion $X{ₙ}^x=MₙX_{n-1}^x+Qₙ+Nₙ\left(X_{n-1}^x\right)$, where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $Nₙ\left(X_{n-1}^x\right)$ is smaller at infinity, than $MₙX_{n-1}^x$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S{ₙ}^x={\sum }_{k=0}^n{X}_k^x$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

A toute mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }(x\wedge x^2)c\left(dx\right)\<\infty$, nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires $(X,H)$ tel que $X$ et $H$ sont indéfiniment divisibles, $H\ge 0$, et pour tout $\lambda \ge 0,E\left(e^{\lambda X}\right)·E\left(e^{-\frac{{\lambda }^2}{2}H}\right)=1$. Plus généralement, à une mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }{e}^{-\alpha x}{x}^{2}c\left(dx\right)\<\infty$ pour tout $\alpha \>{\alpha }_0$, nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent...