Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in \bR \verb2\2 \{0\} (which satisfy some asymptotic dependence conditions), and the corresponding sequence $(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n}, 1\leq j \leq n$. Our main result identifies some explicit conditions for the convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic behavior of the variables in the array

A Cluster Limit Theorem for Infinitely Divisible Point Processes

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of \bR \verb2\2 \{0\}, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences

Functional Convergence of Linear Sequences in a non-Skorokhod Topology

In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form X_i=\sum_{j \in \bZ}c_j \xi_{i-j} with i.i.d. innovations (\xi_i)_{i \in \bZ} and real-valued coefficients (c_j)_{j \in \bZ}. This weak convergence result is obtained in space \bD[0,1] endowed with the $S$-topology introduced in Jakubowski (1992), and the limit process is a linear fractional stable motion (LFSM). One of our result provides an extension of the results of Avram and Taqqu (1992) to the case when the coefficients (c_j)_{j \in \bZ} may not have the same sign. The proof of our result relies on the recent criteria for convergence in Skorokhod's $M_1$-topology (due to Louhichi and Rio (2011)), and a result which connects the weak $S$-convergence of the sum of two processes with the weak $M_1$-convergence of the two individual processes. Finally, we illustrate our results using some examples and computer simulations.Comment: 30 pages, 6 figure

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $\alpha$-stable L\'evy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (1997).Comment: 39 pages; revised version of arxiv 1209.114

A functional central limit theorem for interacting particle systems on transitive graphs

A finite range interacting particle system on a transitive graph is considered. Assuming that the dynamics and the initial measure are invariant, the normalized empirical distribution process converges in distribution to a centered diffusion process. As an application, a central limit theorem for certain hitting times, interpreted as failure times of a coherent system in reliability, is derived.Comment: 35 page

Topological reconstruction of compact supports of dependent stationary random variables

In this paper we extend results on reconstruction of probabilistic supports of random i.i.d variables to supports of dependent stationary $\mathbb R^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the M\"{o}bius Markov chain on the circle is treated at the end with simulations.Comment: 27 pages, 5 figure