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

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 $P=\Big(p_{i,j}\Big)_{i,j}$. 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 $\widetilde{P}$, defined by $\widetilde{P}\Big((u,i),A\times\{j\}\Big)=\mathbb{P}(Y_{n+1}\in A,X_{n+1}=j|Y_n= u,X_n=i)=p_{i,j}F(i,j,A)$, 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 $(i,j)\in E\times E$, $\sqrt{n}\E_i[\exp(\lambda m_n), X_n=j]$ converges to a positive function $H_{i,j}(\lambda)$ and we obtain further informations on this limit function as $\lambda\to 0^+$.