arXiv subjects

Loic Hervé

Publications and source records attributed to Loic Hervé.

2 recordsLinked to original sources

arXiv2019

State-discretization of $V$-geometrically ergodic Markov chains and convergence to the stationary distribution

Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $π$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehatπ_k)_{k\ge 1}$ of probability measures which approximates $π$ as $k$ growths to infinity. The probability measure $\widehatπ_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehatπ_k)_{k\ge 1}$ to $π$ is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for $π$ is known.

math.PR↗

arXiv2008

Stable laws and products of positive random matrices

Let $S$ be the multiplicative semigroup of $q\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\geq1}$ verifies the contraction property $¶(\bigcup_{n\geq1}[X^{(n)} \in S^\circ])>0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix $X_1$ which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain of attraction of a stable law.

math.PR↗