Correction

In my article (M. Rozenblat-Roth, “On the Strong Law of Large Numbers for Nonhomogeneous Markov Chains”), published in DAN, vol. 141, No. 6, 1961, the following corrections must be made.

In Lemma 1 the inequality should read

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

Lemma 2 should read:

Lemma 2. In order that a sequence of random variables \(\xi_i\) \((i\in I)\), connected in a Markov chain, satisfy the strong law of large numbers, it is necessary that, for every \(\varepsilon>0\), the condition

\[ \sum_{n=0}^{\infty} \mathbf{P}\{|\xi_n-\mathbf{M}\xi_n|>\varepsilon n \mid |\xi_{n-1}-\mathbf{M}\xi_{n-1}|\le \varepsilon(n-1)\}<+\infty. \]

be fulfilled.

For \(\alpha_i>\rho>0\) \((i\in I)\), whatever \(\varphi(n)=o(n)\) may be, this condition ceases to be necessary if \(\varepsilon n\) is replaced in it by \(\varphi(n)\).

M. Rozenblat-Roth