### Professor C. D. Fuh, Institute of Statistics, Taiwan

The talks will be based on the following joint papers by Prof. Fuh and Prof. T. L. Lai at Stanford University :

**ASYMPTOTIC EXPANSIONS IN MULTIDIMENSIONAL MARKOV RENEWAL THEORY**- We consider a Markov random walk $ \{(X_n,S_n), n \geq 0 \}$ in which $X_n$ takes values in a general state space and $S_n$ takes values in {\bf R}$^d$, and derive an asymptotic expansion for multidimensional Markov renewal theory. The results yield an asymptotic expansion for the variance of the first passage time $\tau_b= \inf \{n: S_n > b\}$ for $ b>0$, when $S_n$ is a one dimensional Markov random walk with positive drift. The results are also applied to the asymptotic expansions of stopped random walks and products of Markov random matrices.
**POISSON EQUATION, WALD'S IDENTITIES AND QUICK CONVERGENCE FOR MARKOV RANDOM WALKS**- We provide tail probability and moment inequalities, and sufficient conditions for the quick convergence for Markov random walks, without the assumption of uniform or Harris recurrency for the underlying Markov chain.Our approach is based on the Poisson equation and its associated martingale and Wald equation. We also provide Wald equations for the second moment and a variance formula for Markov random walks.
**CORRECTED DIFFUSION APPROXIMATIONS FOR RUIN PROBABILITIES IN MARKOV RANDOM WALKS**- Let $(X,S)=\{(X_n, S_n); n \geq 0\}$ be a Markov random walk with finite state space. For $a \leq 0 <b$ define the stopping times $\tau=\inf\{n:S_n>b\}$ and $T=\inf\{n:S_n \not\in (a,b)\}$. The diffusion approximations of a one-barrier probability P\{\tau<\infty|X_0=i\}$, and a two-barrier probability $P\{S_T\geq b|X_0=i\}$ with correction terms are derived. Furthermore, the limiting distributions of overshoot for a driftless Markov random walk are involved, to approximate the above ruin probabilities.
**A NONLINEAR MARKOV RENEWAL THEORY WITH APPLICATIONS TO SEQUENTIAL ANALYSIS**- Let $T$ be the first time that a perturbed Markov random walk crosses a nonlinear boundary. One concerns the approximations of the distribution of excess over the boundary, the expected stopping time $E_{\nu} T$, where $E_{\nu}$ denotes the expectation under the Markov chain with initial distribution $\nu$. Applications to sequential analysis of hidden Markov chains and random coefficient autoregression models are given.

**Schedule** : Tue. June 1^{st}, 11am-12noon, 2pm-3pm, and Wed. June 2^{nd}, 11am-12noon, 2pm-3pm. Location : 301 Mudd Bldg. Please contact for details.

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