Abstract Generated abstract
This paper studies homogeneous Markov processes with discrete time on an abstract measurable space under recurrence and minorization conditions formulated on a measurable subset. It introduces return kernels to this subset and uses them to derive criteria for the existence of a limiting stationary distribution, including convergence in total variation when the sigma algebra is separable, as well as a complementary null-recurrent type result when the mean return quantity is infinite. The paper also proves a central limit theorem for additive functionals of the process under uniform return and moment conditions, extending Doeblin’s theorem and allowing cases in which the stationary second moment of the function is infinite.
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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).