By Mark A. Pinsky, Samuel Karlin

ISBN-10: 0123814162

ISBN-13: 9780123814166

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**Additional info for An introduction to stochastic modeling**

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1 is recurrent and 1r = 1 is a positive invariant vector. However, 1r Tl is not convergent and so the symmetric random walk is not positive recurrent. It is worth noting that the existence of an invariant vector alone does not imply that {Xn} is recurrent. 4 Finite Markov chains In this section, we assume that the state space is finite and given by N = {0, 1, · · ·, N}. Suppose that a finite Markov chain {Xn} has one recurrent class. 30) where the submatrix Q corresponds to the recurrent class and T the set of transient states.

33) Note that the diagonal elements of P, except the two boundaries, are a 0 = P[Y = 0], that those above (below, respectively) are a 1 (a_l), and so on. This is so because {Xn} is spatially homogeneous with two boundaries. A sufficient condition for aperiodicity is a 0 > 0. 3, let U1 , U2 , · · · be liD positive integervalued random variables representing, for example, the lifetimes of successively replaced systems. Replacements are assumed to be instantaneous. Suppose that the system is replaced at age N + 1 even if the actual lifetime is longer than N + 1.

5 Fora;(n) defined on z+ X Z+, suppose that a;(n) is monotone inn for each i and that L:; a;(n)b;, n E Z+, exist, where b; zs nonnegative. Define a;= limn-+oo a;(n). Ifl:; a;b; exists then 00 00 lim ~ a;(n)b; = ~ a;b;. , a;(O):::; a;(l)::; · · · ::; a;. 6 and the result follows. The other case follows similarly. D = Let Tj be the first passage time to state j and denote by 11-i Ei [Tj], the mean return time to state j. The next theorem is often referred to as the basic limit theorem. Theorem 2.

### An introduction to stochastic modeling by Mark A. Pinsky, Samuel Karlin

by Mark

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