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The nice question below was answered in the affirmative in On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$ (for all $n\geq 1$)?

My question is, can one characterise distributions of $X$ for which this type of monotonicity holds for all $c>\mathbb{E}(X).$

Or maybe it is easier to show for $c$ greater than the median of $X$.

It might be that a nonincreasing probability density is sufficient. But is it necessary?

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  • $\begingroup$ Of course, it is not necessary: one can construct many examples of distributions which satisfy this property, but don't have a non-decreasing density (or even have no density at all). But some assumption is certainly needed. Actually, for each $n$ it is possible to construct a distribution such that $P(S_n > cn) > P(S_{n-1}>c(n-1)) >\dots >P(X_1>c)$ for some $c>EX_1$. $\endgroup$
    – zhoraster
    Commented Mar 16, 2019 at 11:16
  • $\begingroup$ Interesting. Could you elaborate. $\endgroup$
    – kodlu
    Commented Mar 16, 2019 at 11:47
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    $\begingroup$ For a fixed $n$, consider such distribution: $P(X_1 = n) = 1-P(X_1=0) < 1/n$. Then $EX_1<1$ and for $k=1,\dots,n$ the probability $P(S_k >k) = P(\exists i\le k: X_i = n)$ is obviously increasing in $k$. $\endgroup$
    – zhoraster
    Commented Mar 16, 2019 at 18:02

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