Abstract
Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L 2(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap(P) and the convergence rate of P n. When P is transient, the convergence rate of P n is equal to 1 − gap(P). When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of P n in terms of gap(P). These results are extended to L ∞(π)-space.
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Supported in part by 985 Project, 973 Project (Grant No. 2011CB808000), National Natural Science Foundation of China (Grant No. 11131003), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20100003110005) and the Fundamental Research Funds for the Central Universities
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Mao, Y.H., Song, Y.H. Spectral gap and convergence rate for discrete-time Markov chains. Acta. Math. Sin.-English Ser. 29, 1949–1962 (2013). https://doi.org/10.1007/s10114-013-2594-1
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DOI: https://doi.org/10.1007/s10114-013-2594-1