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- W4242000668 abstract "Let { X n , n = 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩 + = {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix 𝐏 = ( p ij ), and let p ij ( n ) = P ∈ X n = j, X k ∈ 𝒩 + for k = 0, 1, ···, n | X 0 = i ], i, j 𝒩 + . The prime concern of this paper is conditions for the existence of the limits, q ij say, of as n → ∞. If the distribution ( q ij ) is called the quasi-stationary distribution of { X n } and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vector x = ( x i ) satisfying r x T = x T 𝐏 and exists, where r is the convergence norm of 𝐏 , i.e. r = R –1 and and T denotes transpose, then it is unique, positive elementwise, and q ij ( n ) necessarily converge to x j as n →∞. Unlike existing results in the literature, our results can be applied even to the R -null and R- transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix is R- transient is discussed to demonstrate the usefulness of our results." @default.
- W4242000668 created "2022-05-12" @default.
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- W4242000668 date "1992-03-01" @default.
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- W4242000668 title "On the existence of quasi-stationary distributions in denumerable R-transient Markov chains" @default.
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- W4242000668 doi "https://doi.org/10.2307/3214788" @default.
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