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Assume that $a_n$, $b_n$ and $c_n$ are 0-1 sequences such that $$ a=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n, \, c=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N c_n, \, d=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n c_n, \, b=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n $$ exist. Suppose also that $a=b$ and $d=ac$. Can I conclude that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n c_n =d $$ ?

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Let:

  • $a_n=1$ if $n\equiv 0,1$ mod $4$
  • $b_n=1$ if $n\equiv 1,3$ mod $4$
  • $c_n=1$ if $n\equiv 0,2$ mod $4$

Then $a=c=\tfrac 12$ and $a=b$ and $d=ac,$ but $\sum_{n=1}^Nb_nc_n$ is always zero.

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