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I wonder if Cesaro summation for matrices is the same as summation for sequences. Cesaro summation for sequences means convergence of the arithmetic means (averages) of partial sums of sequence. For the sequence $a_1, a_2, \ldots$, Cesaro summation means the convergence of $$ \sum_{k=1}^n \frac{\sum_{j=1}^k a_j}{n} $$ when $n\to \infty$

For ergodic Markov chains, the powers of transition matrix $P^n$ are Cesaro summable (e.g., Theorem 5.1.4 in Kemeny, Snell. Finite Markov Chains)

Let us take the simplest transition matrix of an ergodic chain $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$ But the Cesaro sum for $\{P^n \}$, obviously, does not converge. Whereas, $\frac{1}{n+1} \sum_{k=0}^n P^k$ converges to $$ \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} $$ What is the matter?

Konstantin
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The definition of Cesaro summability you give is for that of the series $\ S_n=\sum_\limits{i=1}^na_i\ .$ Kemeny and Snell's theorem $\ 5.1.4\ $ does not say that the series $\ \sum_\limits{i=1}^nP^i\ $ is Cesaro summable. Part (a) says

The $\color{red}{\it{sequence}}$ $\ P^n\ $ is Cesaro-summable to $\ A\ .$

According to the standard definition of Cesaro summability of sequences, as also given by Kemeny and Snell themselves (on p.18 of their first edition of 1960) this means that $$ A=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^nP^i\ . $$ As you've already discovered, for your matrix $\ P\ $ the Cesaro sum of the sequence $\ \big\{P^n\big\}\ $ is $$ A=\pmatrix{\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}}\ , $$ but the series $\ \sum_\limits{i=1}^nP^i\ $ is not Cesaro summable.

Part (b) of Kemeny and Snell's theorem $\ 5.1.4\ $ says

The $\color{red}{\it{series}}$ $\ I+\sum_\limits{i=1}^n\big(P^i-A\big)\ $ is Cesaro-summable to $\ Z\ .$

where $\ Z\ $ has been defined earlier (on p.100 of the first edition) as $\ (I+A-P)^{-1}\ .$ This means that $$ Z=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=1}^n\left(I+\sum_\limits{i=1}^j\big(P^i-A\big)\right)\ ,\tag{1}\label{e1} $$ but says nothing whatever about the Cesaro summability of the series $\ \sum_\limits{i=1}^nP^i\ .$

For your matrix $\ P\ ,$ we have \begin{align} Z&=\pmatrix{\frac{3}{2}&-\frac{1}{2}\\-\frac{1}{2}&\frac{3}{2}}^{-1}\\ &=\pmatrix{\frac{3}{4}&\frac{1}{4}\\\frac{1}{4}&\frac{3}{4}}\ , \end{align} and $$ \sum_{i=1}^{j}\big(P^i-A\big)=\cases{\pmatrix{-\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&-\frac{1}{2}}&if $\ j\ $ is odd\\ \pmatrix{0&0\\0&0}&if $\ j\ $ is even.} $$ You might like to check that the identity (\ref{e1}) does indeed hold for your matrix $\ P\ .$

Note also that the definition of ergodicity for Markov chains given by Kemeny and Snell is much less common nowadays than it was in the $1960$s. The definition used by most modern texts requires a chain to be aperiodic to be classified as ergodic.

lonza leggiera
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  • Such definition of Cesaro summation is given in Wikipedia and, for example, here: https://www.johndcook.com/blog/2020/07/26/cesaro-summation/. First, partial sums are calculated, later the arithmetic means of first $k$ sums. The chain in example is ergodic, cyclic, but not regular – Konstantin Dec 24 '23 at 18:50
  • The definition of Cesaro summability you link to is that for the series $\ S_n=\sum_\limits{i=0}^na_i\ $, not for the sequence $\ \big{a_n}_{n=0}^\infty\ .$ I've now updated my answer in recognition of this confusion. My apologies for not checking Kemeny and Snell's definition of ergodicity for Markov chains. I had forgotten that this definition was much more common in the past. However, it's much less commonly used by more modern texts, which typically require a Markov chain to be aperiodic to be classified as ergodic. – lonza leggiera Dec 24 '23 at 23:17
  • I absolutely agree that a sequence ${P^k}$ is Cesaro summable according to the definition in Kemeny & Snell, p. 18. But there is another definition (Wkipedia etc.), according to which Cesaro summation means the convergence of arithmetic means of partial sums. Isn't it? Thank you for involvement of the fundamental matrix $Z$, but there are no issues with it. – Konstantin Dec 25 '23 at 10:00
  • Since your question concerns the correctness or otherwise of a theorem in Kemeny and Snell's book, it's only their definitions that are relevant in deciding the issue. The definition given in Wikipedia is simply wrong, but this isn't an appropriate place to discuss that. I've briefly described here how a correct definition of the Cesaro sum of a series came to be replaced in that article with an incorrect one of the Cesaro sum of a sequence. – lonza leggiera Dec 28 '23 at 22:12