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arXiv2009

Irregular sets and Central Limit Theorems for dependent triangular arrays

In previous papers, we studied the asymptotic behaviour of $S_N(A,X)=(2N+1)^{-d/2}\sum_{n \in A_N} X_n,$ where $X$ is a centered, stationary and weakly dependent random field, and $A_N=A \cap [-N,N]^d$, $A \subset \mathbb{Z}^d$. This leads to the definition of asymptotically measurable sets, which enjoy the property that $S_N(A;X)$ has a Gaussian weak limit for any $X$ belonging to a certain class. Here we extend this type of results to the case of weakly dependent triangular arrays and present an application of this technique to regression models. Indeed, we prove that CLT and related results hold for $X_n^N=φ(ξ_n^N,Y_n^N), n \in \mathbb{Z}^d$, where $φ$ satisfies certain regularity conditions, $ξ$ and $Y$ are independent random fields, $ξ$ is weakly dependent and $Y$ satisfies some Strong Law of Large Numbers.

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arXiv2009

Estimation of safety areas for epidemic spread

In this work we study safety areas in epidemic spred. The aim of this work is, given the evolution of epidemic at time $t$, find a safety set at time $t+h$. This is, a random set $K_{t+h}$ such that the probability that infection reaches $K_{t+h}$ at time $t+h$ is small. More precisely, inspired on the study of epidemic spread, we consider a model in which the measure $μ_n(A)$ is the incidence -density of infectives individuals- in the set $A$, at time $n$ and $$μ_{n+1}(A)(ω)=\int_S{π_{n+1}(A;s)(ω)μ_n(ds)(ω)}, {for any Borel set} A, $$ with random transition kernels of the form $$π_n(.;.)(ω)=Π(.;.)(ξ_n(ω),Y_n(ω)),$$ where $ξ$, $Y$ satisfy some ergodic conditions. The support of $μ_n$ is called $S_n$. We also assume that $S_0$ is compact with regular border and that for any $x,y$ the kernel $Π(.;.)(x,y)$ has compact support. A random set $K_{n+1}$ is a safety area of level $α$ if: [{$i$)}] $K_{n+1}$ {\rm is a function of} $S_0, S_1, ...,S_n.$ [{$ii$)}] $P(K_{n+1} \cap S_{n+1} \neq \emptyset)\leq α.$ We present a method to find these safety areas and some related results.

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