有限状态马尔可夫链的某些研究

STUDIES IN MARKOFF CHAINS WITH A FINITE NUMBER OF STATES

摘要: 设（ξ_n,n≥0）是状态空间为l,2,…,s的不可约马尔可夫链,其转移概率矩阵是P.用v_ni表示在ξ₁,ξ₂,…,ξ_n中状态i出现的次数（i=1,2,…,s）.用（q₁,…,q₂）表示对应于P的唯一平稳分布.设a₁,…,a_s是满足条件 q₁a₁+q₂a₂+…+q_sa_s=0的任意实数. 在本文中,我们求出了v_n1,…,v_ns的矩母函数,给出了\frac1\sqrtn \sum_1^8 a_i v_n i及（\fracv_n 1-n q_1\sqrtn,…,\left.\fracv_n 8-n q_s\sqrtn\right))的极限分布的明显表达式（当n→∞时）。一些有关的结果也得到了。

Abstract: Let \left(\xi_n, n \geqslant 0\right) be an irreducible Markov Chain on the state-space \1,2, \cdots, s\ with transition probability matrix P. Denote by v_m(i=1, \cdots, s) the number of occurrences of stato i among \xi_1, \cdots, \xi_n. Let \left(q_1, \cdots, q_s\right) be the only stationary distribution for P and a_1, \cdots, a_2 be any real numbers satisfying q_1 a_1+q_2 a_2+\cdots+q_8 a_3=0. In this paper, the gonorating function of moments for v_n 1, \cdots, v_n s and the explicit oxprossions of the limiting distribations for \frac1\sqrtn \sum_1^2 i_i \gamma_a t and \left(\fracv_n 1-n q_1\sqrtn, \cdots, \fracv_n s-n q_2\sqrtn_6\right) are obtained respectively. In addition, several rolated results are given.