ONE-POINT CONNECTION OF MARKOV CHAINS

Let E=\0,1,2, \cdots\, and P(k)=\left(p_y j(k)\right)_i, j \in B, k=1,2, \cdots, be a sequence of transition probability matrices. Suppose that \sum_k=1 \sum_j>1 p_k j(k)< 1. Define \begingathered E_0=\0\ \cup\(k, i): k, i \geqslant 1\ \\ p_(l, b),(k, f)=\delta_k i p_i j(k), l, k, i, j \geqslant 1, \\ p_(l, i), 0=p_i 0(1), p_0,(l, i)=p_0:(1), l, i \geqslant 1 \\ p_00=1-\sum_k \neq 1 \sum_j=1 p_0 j(k). \endgathered Then P=\left(p_\mu \nu\right)_\mu, v \in E_t is called the one-point conneotion of P(k), k \geqslant 1. It is shown that P is irreducible, recurrent or nonrecurrent iff each of P(k), k \geqslant 1, is irreducibly,recurrent or nonrecurrent respectively. In the nonrecurrent case, the probability that the Markov chain tends to infinity along the k-th branch starting from the state 0 is equal to \sum_j=1 p_0 j(k)\left(1-f_j 0(k)\right) / \sum_k>1 \sum_j>1 p_0 j(k)\left(1-f_50(k)\right) Hence, the quantity sup \sum_j \neq i p_i j\left(1-f_j i\right) can be considered as the degree of nonreourrenoe of Markov chains. The one-point connection of a sequence of Q-matrices is discussed in similar manner