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Due to its clustering and self-exciting properties, the Hawkes process has been used extensively in numerous fields ranging from sismology to finance. Since data is often aquired on regular time intervals, we propose a piece-wise constant model based on a Discrete-Time Hawkes Process (DTHP). We prove that this discrete-time model converges to the usual continuous-time Hawkes process as the time-step tends to zero.
In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavins calculus defined on an underlying Poisson embedding.
The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper
A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on $mathbb{Z}^2$ with
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer li
The following type exponential convergence is proved for (non-degenerate or degenerate) McKean-Vlasov SDEs: $$W_2(mu_t,mu_infty)^2 +{rm Ent}(mu_t|mu_infty)le c {rm e}^{-lambda t} minbig{W_2(mu_0, mu_infty)^2,{rm Ent}(mu_0|mu_infty)big}, tge 1,$$ whe