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Asymptotics of the stationary distribution of an oscillating random walk. (Russian, English) Zbl 1061.60047
Sib. Mat. Zh. 45, No. 5, 1112-1129 (2004); translation in Sib. Math. J. 45, No. 5, 915-930 (2004).
Let \(\{\xi_n^{(i)}\}_{n=1}^\infty\), \(i=0,1,2\), be three independent sequences of independent random variables identically distributed within each sequence. Let \({\mathbb E}\xi_n^{(1)}<0\) and \({\mathbb E}\xi_n^{(2)}>0\). Let \(a\) and \(b\), \(a\leq0\leq b\), be arbitrary numbers. The authors consider the random walk \(X_n\) with two levels of switching: \(X_n=X_{n-1}+\xi_n^{(0)}\) if \(X_{n-1}\in[a,b]\), \(X_n=X_{n-1}+\xi_n^{(1)}\) if \(X_{n-1}>b\), and \(X_n=X_{n-1}+\xi_n^{(2)}\) if \(X_{n-1}<a\). The authors find the asymptotics of the stationary distribution of this Markov chain when the distance between the boundaries increases indefinitely, i.e., \(b-a\to\infty\). The article under review is closely related to the above reviewed paper by the same authors (Zbl 1061.60046).
MSC:
60G50 Sums of independent random variables; random walks
60J05 Discrete-time Markov processes on general state spaces
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