We present limit theorems for a sequence of Piecewise Deterministic Markov Processes (PDMPs) taking values in a separable Hilbert space. This class of processes provides a rigorous framework for stochastic spatial models in which discrete random events are globally coupled…

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities…

Suppose that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent and identically distributed (i.i.d) real random variables with $EX_{11}=0$ and $EX_{11}^{2}=1$, and that $\{Y_{jk}, j=1,\cdots,p_2; k=1,\cdots,n\}$ are i.i.d real random variables with $EY_{11}=0$ and $EY_{11}^{2}=1$, and that $\{X_{jk}, j=1,\cdots,p_1; k=1,\cdots,n\}$ are independent of $\{Y_{jk}, j=1,\cdots,p_2;…

Consider a Markov process $\omega_t$ at stationarity and some event $\mathcal{C}$ (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation $\mathbb{P}[\omega_0,\omega_t \in \mathcal{C}] - \mathbb{P}[\omega_0 \in \mathcal{C}]^2$. A second…

Let $\mathcal{T}$ be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The $\lambda$-biased random walk $(X_t)_{t\ge0}$ on $\mathcal{T}$ is the nearest-neighbor random walk which, when at a vertex $v$ with $d_v$ offspring,…

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real  random variables such that $$ \mathbb{E}[a_{j}]=0, \ \ \mathbb{E} [|a_{j}|^{2}]=1 \ \mathrm{for}\,\ \ j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty\…

We consider population models in which the individuals reproduce, die and also migrate in space. The population size scales according to some parameter $N$, which can have different interpretations depending on the context. Each individual is assigned a mass of $1/N$ and…