# Monotonicity of $\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c \right]$ in $n$ for $c>\mathbb{E}(X).$

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?

• 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 $c>EX_1$. – zhoraster Mar 16 at 11:16
• Interesting. Could you elaborate. – kodlu Mar 16 at 11:47
• 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 $k$. – zhoraster Mar 16 at 18:02