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Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a sequence $c_n>0$ such that $$ \sum_{n=1}^\infty c_n \xi_n \quad \text{converges almost surely.} $$

Can one give an example to show that the "almost surely" in the statement CANNOT be strengthened to "pointwise"?

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    $\begingroup$ RVs in general are not defined pointwise EVERYWHERE, so this question does not make a lot of sense. $\endgroup$
    – Asaf
    Commented Oct 5, 2016 at 20:21
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    $\begingroup$ To make sense of the question, you might say: find a sequence of real-valued Borel functions $\xi_n$ on, say, $[0,1]$, such that there is no sequence $c_n > 0$ for which $\sum_{n=1}^\infty c_n \xi_n$ converges pointwise. $\endgroup$
    – Robert Israel
    Commented Oct 5, 2016 at 20:24

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Let $\Omega=(0,1)$ and $\xi_n:\omega\in\Omega\mapsto$ the $n$-th term of the continued fraction expansion of $\omega$. Given a sequence $c_n$, there is another sequence $m_n\in\mathbb{N}$ such that $\sum_{n=1}^\infty c_nm_n$ diverges. Let $x=[m_1,m_2,\dots,m_n,\dots]$. Then $\sum_{n=1}^\infty c_n\xi_n(x)$ diverges.

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