Limit of the Cesaro sum of the product of 0-1 sequences.

Question

Assume that $a_n$ and $b_n$ are 0-1 sequences such that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = p. $$ Let also $c_n$ an other 0-1 sequence. Is it true that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n . $$ I think it is true because if the first equation is correct, than the limit is also true on each subsequence and in particular in the one where $c_n=1$. Is my argument correct?

Answer 1

No, the claimed property does not hold. Take $a_n=1$ iff $n$ is even and $b_n=1$ iff $n$ is odd. Then $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = \frac{1}{2}.$$ Now if $c_n=a_n$ then $a_nc_n=a_n$ whereas $b_nc_n=0$ and it follows that $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n =\frac{1}{2}\not=0 =\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n .$$

Answer 2

It's not true, let consider

• $a_n=0,1,0,1,...$
• $b_n=1,0,1,0,...$
• $c_n=a_n$