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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
##### Keywords:
stationary distribution; factorization method
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