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