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Consider a continuous random variable $X$ with the compact support $[0,1]$. For given $N\in\mathbb{N}$, we define the weighted sum as $$ S_N=\sum_{i=1}^N a_iU_i, $$ where $U_i$ are i.i.d. random variables uniformly distributed in $[0,1]$ and $a_i$ are free parameters.

My question is how well we can approximate $X$ by $S_N$ in distribution. Furthermore, I note that, in spite of the metric for approximation, the approximation accuracy will increase as $N$ increases since we can let some $a_i=0$. Hence, I also want to know the behavior of the best approximation when $N\to \infty$.

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In general, possibly not at all.

Indeed, without loss of generality $a_i\ne0$ for all $i$. Then the pdf of each $a_iU_i$ is log concave and hence (by the well-known Proposition 3.5) the pdf of $S_N$ is log concave.

So, if the pdf of $X$ is substantially not log concave (say U-shaped), then $X$ cannot be approximated by $S_N$ in distribution, even with a however large $N$.

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  • $\begingroup$ @losifPinelis Thanks. We further assume that $S$ is a given log-concave distribution. How to approximate $S$ by the weighted sum of i.i.d. uniform random variables and find the optimal weighting factors? Of course, the asymptotic analysis is welcome. $\endgroup$
    – RyanChan
    Commented Jun 11, 2021 at 7:40
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    $\begingroup$ @RyanChen : I think that, even if the additional log-concavity condition is imposed, there will be no approximation in general. However, to prove that, one will probably need to use some properties of the convolution of uniform distributions that are much less trivial than the log-concavity. So, I believe the question with the additional log-concavity condition should be posted separately, especially given that your posted question has been answered. $\endgroup$
    – Iosif Pinelis
    Commented Jun 11, 2021 at 14:07
  • $\begingroup$ @losifPinelis Thanks for your comment. I will post a new question separately. $\endgroup$
    – RyanChan
    Commented Jun 12, 2021 at 0:08

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