S. V. NAGAEV

SOME QUESTIONS IN THE THEORY OF HOMOGENEOUS MARKOV PROCESSES WITH DISCRETE TIME

(Presented by Academician A. N. Kolmogorov, 23 II 1961)

Let \(\xi_n\) be a Markov process with discrete time. The random variables \(\xi_n\) take values in some abstract space \(X\), on which a \(\sigma\)-algebra \(F_X\) is defined. Denote the transition probability in \(n\) steps by \(p^{(n)}(x,B)\), \(x \in X\), \(B \in F_X\).

Suppose that the process \(\xi_n\) satisfies the following conditions:

A. There exist \(A \in F_X\), a finite measure \(\varphi(\cdot)\) on \(AF_X\) with \(\varphi(A) > 0\), \(k_0 \geq 1\), and \(\delta > 0\) such that

\[ p_0^{(k_0)}(x,y) \geq \delta > 0 \]

for \(x \in A\), \(y \in A\), where \(AF_X\) is the \(\sigma\)-algebra on \(A\) consisting of sets \(AB\), \(B \in F_X\); \(p_0^{(k_0)}(x,y)\) is the density, with respect to \(\varphi(\cdot)\), of the absolutely continuous component of \(p^{(k_0)}(x,\cdot)\).

B.

\[ \operatorname{Pr}\{\xi_n \in X-A,\ n=k_0,2k_0,\ldots \mid \xi_0=x\}=0 \]

for all \(x \in X-A\).

Conditions A, B are more effective than Doeblin’s conditions \({}^{(2)}\), and provide a greater analogy with the case of countable Markov chains \({}^{(1)}\).

Let \(\nu\) be the first of the numbers \(nk_0\), \(n>0\), for which \(\xi_{nk_0}\in A\). Put

\[ q_A^{(n)}(x,C)=\mathbf{P}\{\xi_\nu\in C,\ \nu=nk_0\mid \xi_0=x\}, \]

\[ q_A(x,C)=\sum_{n=1}^{\infty} q_A^{(n)}(x,C), \]

\[ \mu_A(x,C)=\sum_{n=1}^{\infty} n q_A^{(n)}(x,C). \]

Let \(q_A(\cdot)\) be the stationary distribution corresponding to the transition function \(q_A(x,C)\), \(x \in A\), \(C \in AF_X\), whose existence follows from condition A.

Theorem 1. If

\[ \int_A \mu_A(x,A)\,q_A(dx)<\infty, \]

then there exists a stationary distribution \(p(\cdot)\) such that

\[ \lim_{n\to\infty} p^{(n)}(x,B)=p(B) \]

for any \(B \in F_X\).

It can be shown that condition A is necessary for Theorem 1 to hold, if \(F_X\) is separable.

Theorem 2. If

\[ \int_A \mu_A(x,A)\,q_A(dx)=\infty, \]

then there exists an increasing sequence \(E_m\in F_X\) such that

\[ \lim_{m\to\infty}E_m=X, \]

\[ \lim_{n\to\infty}p^{(n)}(x,E_j)=0,\qquad j=1,2,\ldots, \]

\[ \lim_{n\to\infty}\sum_{k=1}^{n}p^{(k)}(x,E_m)=\infty, \]

\[ \lim_{n\to\infty}\left[\sum_{k=1}^{n}p^{(k)}(x,E_m)\Big/\sum_{k=1}^{n}p^{(k)}(y,E_m)\right]=1 \]

for any \(x,y\).

Theorem 3. If the conditions of Theorem 1 are satisfied and \(F_X\) is separable, then

\[ \lim_{n\to\infty}\sup_{B\in F_X}\left|p^{(n)}(x,B)-p(B)\right|=0. \]

Let \(f(x)\) be a real function measurable with respect to \(F_X\). Put

\[ M_{k_0}(x,C)= \mathbf{M}\left\{\sum_{k=1}^{\nu} f(\xi_k)\mid \xi_0=x,\ \xi_\nu\in C\right\}q(x,C), \]

\[ M_{k_0}=\int_A M_{k_0}(x,A)\,q_A(dx), \]

\[ B_{k_0}(x)= \mathbf{M}\left\{\left[\sum_{k=1}^{\nu} f(\xi_k)\right]^2\mid \xi_0=x\right\}, \]

\[ \overline{B}_{k_0}(x)= \mathbf{M}\left\{\left[\sum_{k=1}^{\nu}\bigl(f(\xi_k)-M_{k_0}\bigr)\right]^2\mid \xi_0=x\right\}. \]

Let

\[ \sigma_{k_0}^2 = \int_A \overline{B}_{k_0}(x)\,q_A(dx) + 2\sum_{k=1}^{\infty}\int_A q_A(dx_1)\int_A \bigl[M_{k_0}(x_1,dx_2) \]

\[ - M_{k_0}q_A(x_1,dx_2)\bigr] \int_A\bigl[M_{k_0}(x_3,A)-M_{k_0}\bigr]q_A^{(k)}(x_2,dx_3). \]

Theorem 4. If

\[ \sum_{n=1}^{\infty} n q_A^{(n)}(x,A) \]

converges uniformly on \(A\),

\[ \lim_{y\to\infty}\sup_{x\in A}\mathbf{P}\left(|f(\xi_j)|>y\mid \xi_0=x\right)=0,\qquad j=1,2,\ldots, \]

\[ \int_A B_{k_0}(x)\,q_A(dx)<\infty \]

and \(\sigma_{k_0}^2 > 0\), then for any initial distribution

\[ \lim_{n \to \infty} \Pr \left\{ \frac{1}{\sigma_{k_0}}\sqrt{\frac{k_0}{n}} \sum_{k=1}^{n} [f(\xi_k)-M_{k_0}] < u \right\} = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{u} e^{-\lambda^2/2}\,d\lambda . \]

Theorem 4 is a generalization of the well-known theorem of Doeblin \(\left({}^{3}\right)\). Let us note that \(\sigma_{k_0}^2\) may exist also in the case when

\[ \int_X f^2(x)\,p(dx)=\infty . \]

Already by virtue of this circumstance alone, Theorem 4 cannot be regarded as a consequence of the central limit theorem for stationary processes, since the latter is proved under the assumption that the variance is finite \(\left({}^{4,5}\right)\).

V. I. Romanovskii Institute of Mathematics
Academy of Sciences of the Uzbek SSR

Received
23 II 1961

REFERENCES

\({}^{1}\) A. N. Kolmogorov, Matem. sborn., 1 (43), 607 (1936).
\({}^{2}\) W. Doeblin, Ann. Sci. École Norm. Sup. (3), 37 (1940).
\({}^{3}\) W. Doeblin, Bull. Soc. Math. France, 66, 210 (1938).
\({}^{4}\) M. Rosenblatt, Proc. Nat. Acad. Sci. Wash., 42, No. 1 (1956).
\({}^{5}\) I. A. Ibragimov, DAN, 125, No. 4 (1959).