Estimating expectation of a slightly strange sum

Question

Let $X$ be a random variable with support on the positive integers (you can assume $\mathbb{E}[X^2] <\infty$ if needed, or even higher moments if needed), and let $S(i)=\mathbb{P}(X\geq I)$. Consider the random sum $$W =\sum_{i=0}^X S(i)$$

Reasoning heuristically, we're summing about $\mathbb{E}[X]$ terms, the first of which are close to one and the last of which are about $1/2$. So the sum itself should be a bit less than $\mathbb{E}[X]$, but still of the same order. Is this true?

Being a bit more formal, what I'd like to be true is that asymptotically, as $\mathbb{E}[X] \to \infty$, $\frac{\mathbb{E}[W]}{\mathbb{E}[X]}$ stays bounded away from 0. (That it is bounded above by one is trivial.)

Answer 1

It is possible to take random variables $X_k$ and the corresponding $W_k$ so $\mathbb E[X_k] \to \infty$ while $\mathbb E[W_k]/\mathbb E[X_k] \to 0$. For example, consider $X = N$ with probability $p$, $0$ with probability $1-p$. Then $\mathbb E[W] = \sum_{i=0}^\infty S(i)^2 = 1 + N p^2$ while $\mathbb E[X] = Np$, so $\mathbb E[W]/\mathbb E[X] = p + 1/(Np)$. Take $N_k \to \infty$ and $p_k \to 0$ so $N_k p_k \to \infty$ (e.g. $p_k = 1/\sqrt{N_k}$).