Given a real number $\alpha \in [0.5, 1.5]$, an integer number $k>1$, and a set of independent Bernoulli random variables $x_1, \dots, x_n$, I am interested to find a lower-bound for $F(\alpha, k)= \frac{E[\min(X, k)]}{k}$ subject to $E[X] = \alpha k$ where $X=\sum_{i\in n} x_i$.

My observation is that for $k\rightarrow \infty$ we have $F(\alpha, k) = \min(\alpha, 1)$. I can also find a tight lower-bound for k=2, which is $1-\frac{(2+2\alpha)e^{-2\alpha}}{2}$ when $x_i$'s are identical and $n\rightarrow \infty$.

My question is that, is it possible to show that for any $k>2$, the same bound holds? More precisely, I want to show that for any $\alpha$ and $k$, we have $$F(\alpha, k)\geq 1-\frac{(2+2\alpha)e^{-2\alpha}}{2}.$$

  • $\begingroup$ Shouldn't that be $e^{-2\alpha}$? $\endgroup$ – Will Sawin Jan 3 at 22:02

Let $x_1,\dots, x_n$ be independent Bernoulli random variables with expectations $p_1,\dots, p_n$ summing to $k\alpha$. Let $y_1,\dots, y_n$ be independent random variables with expectations $p_1 \frac{k-1}{k},\dots, p_n \frac{k-1}{k}$, summing to $(k-1)\alpha$. I claim that $$ \mathbb E \left[ \frac{ \min \left( \sum_{i=1}^n x_i, k \right)}{k} \right]\geq \mathbb E \left[ \frac{ \min \left( \sum_{i=1}^n y_i, k-1 \right)}{k-1} \right]$$

This implies that the optimal lower bound for a given $k,\alpha$ is an increasing function of $k$, so your $k=2$ lower bound works for all $k$.

To do this, we can couple $x_i$ with $y_i$ so that, whenever $x_i=0$, we also have $y_i=0$, and when $x_i=1$, $y_i$ has a $\frac{k-1}{k}$ conditional probability of being $1$. In other words the conditional expectation of $y_i$ on a given value of $x_i$ is equal to the $\frac{k-1}{k} x_i$. This implies that for $F$ any convex function, $$ \mathbb E \left[ F \left( \frac{k-1}{k} x_1,\dots, \frac{k-1}{k} x_n \right) \right] \geq \mathbb E \left[ F \left( y_1,\dots,y_n \right) \right]$$ by (a conditional form of) Jensen's inequality.

Taking $$F(y_1,\dots,y_n) = \min ( \sum_{i=1}^n \frac{y_i}{k-1}, 1),$$ which is convex, gives our desired inequality.

In what also may be helpful, here is the sharp lower bound for fixed $k>2$:

Given a Bernoulli random variable $x_i$ with mean (probability of being $1$) $p_i$, we can find a Poisson random variable $y_i$ coupled with it, with the same mean, such that if $x_i=0$ then $y_i=0$. Indeed this is just saying that $P(x_i=0) < P(y_i=0)$ which follow from $1-p< e^{-p}$.

Thus we have the conditional probability $\mathbb E[ y_i | x_i=x] =x$. Thus, conditional on $x_1,\dots, x_n$ taking the values $x_1',\dots, x_n'$, the expected value of $\sum_{i=1}^{n} y_i$ is $\sum_{i=1}^n x_i'$, so the expectation of $\min( \sum_{i=1}^{n} y_i,k)$ is at most $\min( \sum_{i=1}^n x_i', k)$. Thus $$\mathbb E \left[ \min\left( \sum_{i=1}^{n} y_i, k\right)\right] \leq \mathbb E \left[ \min\left( \sum_{i=1}^{n} x_i, k\right)\right]. $$

But $\sum_{i=1}^{n} y_i$ is a Poisson random variable with known distribution, so we get a lower bound of

$$ \sum_{j=0}^{\infty} \frac{ \min(j,k)}{k} \frac{ (k\alpha)^j e^{-k \alpha} }{ j!}$$

When $k=2$ this is exactly your stated bound.

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  • $\begingroup$ Thanks for the answer. Could you please explain a little bit more about how you use Jensen's inequality? $\endgroup$ – Melika Jan 7 at 22:54
  • $\begingroup$ @Melika For each possible value of the $x_i$, we apply Jensen's inequality to the expectation of $F(y_1,\dots,y_n)$ conditional on the $x_i$ having that value. This gives a lower bound of $F( \frac{k-1}{k} x_1,\dots, \frac{k-1}{k} x_n)$. Then we average over the possible values of the $x_i$. $\endgroup$ – Will Sawin Jan 7 at 23:19

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