Bound on Expectation and Variance of Random variables ratio with positive weights

Question

Let $X_1, X_2, \dots, X_n$ be $n$ strictly positive iid random variables with bounded variance. Let $w_1, w_2, \dots, w_n$ be non-negative deterministic constants such that $\sum_{i=1}^{n} w_i = 1$. Consider the random variable $$Y_j = \frac{w_jX_j}{\sum_{i=1}^{n} w_i X_i}$$

What are the upper and lower bounds on the expectation and variance of $Y_j$? An easy upperbound on variance can be established by Popoviciu's inequality, but I am looking for bounds that use moment information of $X_i$'s.

For the case where $w_i = \frac{1}{n}$, it is easy to show that the $E(Y_j) = \frac{1}{n}$ and have been asked numerous times here (see, e.g. this question). For the general case, there are counterexamples to show $E(Y_j) \neq w_j$ in general (see this question). However, based on my observations the expectation is very close to $w_j$, at least for the case where $X_i = \alpha+$Bernoulli($p$) for some deterministic constant $\alpha>0$.

Answer 1

Considering that

$$\frac{w_j}{\max_{i=1,...,n}\{w_i\}} \frac{X_j}{\sum_{i=1}^{n} X_i} \le Y_j = \frac{w_jX_j}{\sum_{i=1}^{n} w_i X_i}\le \frac{w_j}{\min_{i=1,...,n}\{w_i\}} \frac{X_j}{\sum_{i=1}^{n} X_i}$$

one can obtain

$$\frac{1}{n}\frac{w_j}{\max_{i=1,...,n}\{w_i\}} \le \mathbb{E}(Y_j) \le \frac{1}{n}\frac{w_j}{\min_{i=1,...,n}\{w_i\}}. $$

The bounds are tight for the classical case that all $w_j$ are the same, the upper and lower bounds become equal and $\mathbb{E}(Y_j)=\frac{1}{n}.$

I am not sure whether these bounds are also tight when $w_j$ are arbitrary and fixed.

Answer 2

In general, if $X$ and $Y$ are positive and independent we have $$E\left(\frac{X}{X+Y}\right)=\int_0^{\infty}E(Xe^{-sX})E(e^{-sY})ds=-\int_0^{\infty}\frac{d}{ds}E(e^{-sX})E(e^{-sY})ds,$$ a fact that you can apply to $X=w_1X_1$ and $Y=w_2X_2+\cdots+w_nX_n.$ If $X\sim \alpha +B$ with $B$ Bernoulli, the integral is computable for small values of $n.$