A Local Limit Theorem for the Minimum of a Random Walk with Markovian Increasements

Abstract

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space and $E$ be a finite set. Assume that $X=(X_n)$ is an irreducible and aperiodic Markov chain, defined on $(\Omega ,\mathcal{F},\mathbb{P})$, with values in $E$ and with transition probability $P=(p_{i,j}{)}_{i,j}$. Let $(F(i,j,\d x))_{i,j\in E}$ be a family of probability measures on $\mathbb{R}$. Consider a semi-markovian chain $(Y_n,X_n)$ on $\mathbb{R}\times E$ with transition probability $\tilde{P}$, defined by $\tilde{P}((u,i),A\times \{j\})=\mathbb{P}(Y_{n+1}\in A,X_{n+1}=j|Y_n=u,X_n=i)=p_{i,j}F(i,j,A)$, for any $(u,i)\in \mathbb{R}\times E$, any Borel set $A\subset \mathbb{R}$ and any $j\in E$. We study the asymptotic behavior of the sequence of Laplace transforms of $(X_n,m_n)$, where $m_n=\min (S_0,S_1,...,S_n)$ and $S_n=Y_0+...+Y_{n-1}$. Under quite general assumptions on $F(i,j,dx)$, we prove that for all $(i,j)\in E\times E$, $\sqrt{n}\E_i[\exp(λm_n), X_n=j]$ converges to a positive function $H_{i,j}(\lambda )$ and we obtain further informations on this limit function as $\lambda \to 0^{+}$.

arXiv comment40 pages, 3 figures; updated author's present address, corrected typos, unified notations