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If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. I'd like to go further and say that it is dramatically false. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that a randomly generated sequence of 0's and 1's almost never have property $(*)$ ; but almost every such sequence is Cesaro summable to 1/2 by the law of large numbers.

If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. If each $x_n$ is chosen uniformly at random from $\{0,1\}$ then this sequence almost never has property $(*)$ (see Connor's appropriately named article Almost none of the sequences of 0's and 1's are almost convergent)

However the Cesaro limit of this random sequence $(x_n)$ is almost always $1/2$ by the law of large numbers.

If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. I'd like to go further and say that it is dramatically false. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that randomly generated sequence of 0's and 1's almost never have property $(*)$ ; but almost every randomly generated sequence of 0's and 1's is Cesaro summable to 1/2 by the law of large numbers.

If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. I'd like to go further and say that it is dramatically false. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that a randomly generated sequence of 0's and 1's almost never have property $(*)$ ; but almost every such sequence is Cesaro summable to 1/2 by the law of large numbers.

If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that randomly generated sequence of 0's and 1's almost never has property $(*)$ ; but almost every randomly generated sequence of 0's and 1's is Cesaro summable to 1/2 by the law of large numbers.

If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff $$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L \quad (*) $$ uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. I'd like to go further and say that it is dramatically false. It was shown by Connor (in his appropriately named article) Almost none of the sequences of 0's and 1's are almost convergent that randomly generated sequence of 0's and 1's almost never have property $(*)$ ; but almost every randomly generated sequence of 0's and 1's is Cesaro summable to 1/2 by the law of large numbers.