Doklady of the Academy of Sciences of the USSR
1962. Volume 147, No. 6

MATHEMATICS

M. ROSENBLATT-ROT

ON THE LAW OF LARGE NUMBERS AND ON THE STRONG LAW OF LARGE NUMBERS FOR NONHOMOGENEOUS MARKOV CHAINS

(Presented by Academician A. N. Kolmogorov on June 27, 1962)

Let \(\alpha_i\) be the coefficient of ergodicity of the \(i\)-th probability transition function of a certain Markov chain, nonhomogeneous in general (see \((^{1,2})\)), and denote
\[ \alpha^{(n)}=\min_{1\le i\le n}\alpha_i\quad (n\in I;\ I=(1,2,\ldots)). \]
If \(l\) is some natural number \((l>1)\), let
\[ \sigma_m^{-1}=l^{2m}\alpha(l^{m+1}),\qquad \omega_i^{-2}=\sum_{m=\rho}^{\infty}\sigma_m, \]
where \(\rho=[\log i/\log l]\).

Theorem 1. In order that a sequence of random variables \(\xi_i\ (i\in I)\), connected into a nonhomogeneous Markov chain for which \(\alpha^{(n)}>0\) \((n\in I)\), \(n\alpha^{(n)}\to\infty\) \((n\to\infty)\), and depending on \(r\ge 1\) moments of time, obey the law of large numbers, it is sufficient that
\[ \lim_{n\to\infty}\frac{1}{n^2\alpha^{(n)}}\sum_{i=1}^{n}\mathbf D\xi_i=0. \]

Theorem 2. If a sequence of random variables \(\xi_i\ (i\in I)\), connected into a nonhomogeneous Markov chain with \(\alpha^{(n)}>0\) \((n\in I)\), \(n\alpha^{(n)}\to\infty\) \((n\to\infty)\), satisfies the condition
\[ \sum_{n=1}^{\infty}\frac{\mathbf D\xi_n}{\omega_n^2}<\infty, \]
then it obeys the strong law of large numbers.

If
\[ \sigma_{m+1}<k\sigma_m\quad (k<1;\ m\ge \rho), \]
this condition takes the form
\[ \sum_{n=1}^{\infty}\frac{\mathbf D\xi_n}{n^2\alpha^{(n)}}<\infty. \]

These results are obtained with the aid of several lemmas.

Lemma 1. If the random variables \(\xi_i\ (i\in I)\) are connected into a nonhomogeneous Markov chain with \(\alpha^{(n)}>0\) \((n\in I)\) and depend on \(r\ge 1\) moments of time, then
\[ \mathbf D\left(\sum_{k=1}^{n}\right)\xi_k\le \frac{C}{\alpha^{(n)}}\sum_{k=1}^{n}\mathbf D\xi_k. \]

Lemma 2. If the random variables \(\xi_i\ (i\in I)\) are connected into a nonhomogeneous Markov chain with \(\alpha^{(n)}>0\) \((n\in I)\), then for any \(\varepsilon>0\) the following holds:

inequality

\[ \mathbf P\left\{\max_{1\leqslant s\leqslant n}\left|\sum_{k=1}^{s}(\xi_k-\mathbf M\xi_k)\right|>\varepsilon\right\} \leqslant \frac{1}{\varepsilon_1^2\alpha^{(n)}}\sum_{i=1}^{n}\mathbf D\xi_i, \]

where \(\varepsilon_1\) is a certain positive number depending on \(\varepsilon\) and on the chain.

Lemma 3. Let the random variables \(\xi_i\) \((i\in I)\) be associated with an inhomogeneous Markov chain with \(\alpha^{(n)}>0\) \((n\in I)\); if \(c_i>0\), \(c_i\geqslant c_{i+1}\) \((i\in I)\), \(1\leqslant n\leqslant m\), \(\varepsilon>0\), then the inequality

\[ \mathbf P\left\{\max_{n\leqslant k\leqslant m} c_k\left|\sum_{s=1}^{k}(\xi_s-\mathbf M\xi_s)\right|\geqslant\varepsilon\right\} \leqslant \frac{1}{\varepsilon_1^2\alpha^{(m)}}\left\{ c_n^2\sum_{k=1}^{n}\mathbf D\xi_k+ \sum_{k=n+1}^{m}c_k^2\mathbf D\xi_k \right\}, \]

where \(\varepsilon_1\) is a certain positive number depending on \(\varepsilon\) and on the chain.

For \(\alpha^{(n)}=O(n^{-\beta})\) \((0\leqslant\beta<1)\), Theorem 1 yields Theorem 1 of \({}^{(3)}\); from Theorem 2, if one takes \(k=l^{2-2\beta}<1\), one obtains Theorem 1 of \({}^{(4)}\); from Lemma 2—Lemma 1 of \({}^{(4)}\), and from Lemma 3—Theorem 1 of \({}^{(5)}\).

We note that for \(\alpha^{(n)}\equiv 1\) \((n\in I)\), i.e. for independent random variables, Theorem 1 implies the result of A. A. Markov, while Theorem 2 and Lemma 2 give the results of A. N. Kolmogorov \({}^{(6-9)}\), and Lemma 3 gives the result of Hájek—Rényi \({}^{(10)}\).

In view of the fact that, for arbitrarily dependent random variables, the inequality

\[ \mathbf D\left(\sum_{k=1}^{n}\xi_k\right)\leqslant n\sum_{k=1}^{n}\mathbf D\xi_k \]

holds, the results contained in Theorems 1 and 2 are of interest only when \(n\alpha^{(n)}\to\infty\) \((n\to\infty)\), because otherwise, applying this inequality instead of the inequality of Lemma 1, we obtain better results. Under these conditions, if one assumes that \(n\alpha^{(n)}\) is a monotone sequence, we obtain that \(\sigma_m\) is a nonincreasing sequence, so that the requirement imposed on it in Theorem 2 is not a very strong restriction. We note that this requirement is satisfied for \(\alpha^{(n)}=(\log n)^{-\beta}\) \((\beta>0)\); \(\alpha^{(n)}=(\log_p n)^{-\beta}\) \((p>1)\) \((\log_p n=\log(\log_{p-1}n),\ 0<\beta\leqslant1)\); \(\alpha^{(n)}=n^{-\beta}(\log n)^{-\gamma}\) \((\beta+\gamma<2,\ \beta>0,\ \gamma>0)\); \(\alpha^{(n)}=n^{-\beta}\) \((0\leqslant\beta<1)\).

Faculty of Mathematics and Physics
K. I. Parhon University
Bucharest, Romania

Received
24 III 1962

CITED LITERATURE

\({}^{1}\) R. L. Dobrushin, Theory of Probability and Its Applications, 1, issue 1, 72 (1956).
\({}^{2}\) R. L. Dobrushin, Theory of Probability and Its Applications, 1, issue 4, 365 (1956).
\({}^{3}\) M. Rosenblatt-Roth, DAN, 134, No. 2, 278 (1960).
\({}^{4}\) M. Rosenblatt-Roth, DAN, 141, No. 6, 1310 (1961).
\({}^{5}\) M. Rosenblatt-Roth, C. R., in press.
\({}^{6}\) A. Kolmogoroff, Math. Ann., 99, 309 (1928).
\({}^{7}\) A. Kolmogoroff, Math. Ann., 102, 484 (1929).
\({}^{8}\) A. Kolmogoroff, C. R., 191, 910 (1930).
\({}^{9}\) A. N. Kolmogorov, Basic Concepts of Probability Theory, Moscow—Leningrad, 1936.
\({}^{10}\) J. Hájek, A. Rényi, Acta Math. Acad. Sci. Hung., 6, No. 3—4, 281 (1955).