Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications
Asymptotic behaviour of stationary distributions for countable Markov chains, with some applications
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可数马尔可夫链平稳分布的渐近行为及其一些应用
DOI：
10.2307/3318715
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发表时间：
1999
期刊：
影响因子：
1.5
通讯作者：
R. Iasnogorodski
中科院分区：
文献类型：
--
作者：
S. Aspandiiarov;R. Iasnogorodski
设{Zn，n0}是取值于RD的可数无界子集S上的非周期不可约回归(不一定是正回归)马氏链，R(-)是它的不变测度，f是定义在S上的非负函数.我们首先得到了Jf(Z)r(Dz)=0的充分条件(Twedie得到了Jf(Z)7r(Dz)的相应结果).然后得到了S的子集B上不变测度n的值的上下界，即XR(B)。这些界被表示为首次通过概率和首次离开B的时间。我们还展示了如何利用下界或上鞅技巧来估计后者的量。最后给出了Z2上无漂移反射随机游动和非负实数上具有渐近小漂移的Lamperti型马氏链的结果。在这两种情况下，我们都得到了关于它们的平稳测度的渐近行为的非常精确的信息。
Let { Zn, n 0} be an aperiodic irreducible recurrent (not necessarily positive recurrent) Markov chain taking values on a countable unbounded subset S of Rd, .r(-) its invariant measure and f is a non-negative function defined on S. We first find sufficient conditions under which J f(z)r(dz) = 0 (the corresponding result for the finiteness of J f(z)7r(dz) was obtained by Tweedie). Then we obtain lower and upper bounds for the values of the invariant measure n on the subsets B of S, that is, xr(B). These bounds are expressed in terms of first passage probabilities and the first exit time from B. We also show how to estimate the latter quantities using subor supermartingale techniques. The results are finally illustrated for driftless reflected random walks in Z2 and for Markov chains on nonnegative reals with asymptotically small drift of Lamperti type. In both cases we obtain very precise information on the asymptotic behaviour of their stationary measures.