Cesaro summation for matrices

Question

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?

Answer 1

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.