Talks on Markov Renewal Processes and Change-Point Detection
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 1st, 11am-12noon, 2pm-3pm, and
Wed. June 2nd, 11am-12noon, 2pm-3pm. Location : 301 Mudd
Bldg. Please contact Steve Kou
for details.
For general CAP inquiries: chris@wald.stat.columbia.edu
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Heyde, Director of CAP)
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(Karl Sigman, Secretary of CAP)
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