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The following problem arose in the process of showing the convergence of a particular regression algorithms.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Suppose that $X,X_1,X_2,...:\Omega\to(0,+\infty)$ are $\mathbb{P}-$i.i.d. random variables such that $\forall r>0, \mathbb{P}(X<r)>0$.

Let $r>0$. Is it true that \begin{equation} \mathbb{E}\left[\sum_{k=1}^m \frac{\frac{1}{X_k}}{\sum_{j=1}^m \frac{1}{X_j}}\chi_{(r,+\infty)}(X_k)\right]\to0,m\to\infty? \end{equation} If so, can we somehow specify the speed of convergence in terms of the quantity $m\mathbb{P}_X\big((0,r]\big)$?

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  • $\begingroup$ I removed the "geometric measure theory" tag $\endgroup$
    – T_M
    May 29 '20 at 11:36
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It is false in general.

For example, get $X$ such that $1/X \in L^1$ and $\mathbb{P}(X> r)>0$.

Then by the strong law of large numbers \begin{align} \frac{1}{m} \sum_{j=1}^{m-1} \frac{1}{X_j} \to \mathbb{E}\Big[\frac{1}{X}\Big], m\to \infty, \mathbb{P} \text{-a.e.} \end{align} so, using Fatou's lemma: \begin{align} \operatorname{liminf}_{m \to \infty}\mathbb{E}\bigg[\sum_{k=1}^m \frac{\frac{1}{X_k}}{\sum_{j=1}^m \frac{1}{X_j}}\chi_{(r,+\infty)}(X_k)\bigg] &= \operatorname{liminf}_{m \to \infty} m \mathbb{E}\bigg[\frac{\chi_{(r,+\infty)}(X)}{1+\sum_{j=1}^{m-1}\frac{X}{X_j}}\bigg] \\ &= \operatorname{liminf}_{m \to \infty} \mathbb{E}\bigg[\frac{\chi_{(r,+\infty)}(X)}{\frac{1}{m}+X\frac{1}{m}\sum_{j=1}^{m-1}\frac{1}{X_j}}\bigg] \\ &\ge \mathbb{E}\bigg[ \operatorname{liminf}_{m \to \infty}\frac{\chi_{(r,+\infty)}(X)}{\frac{1}{m}+X\frac{1}{m}\sum_{j=1}^{m-1}\frac{1}{X_j}}\bigg] \\ &= \mathbb{E}\bigg[\frac{\chi_{(r,+\infty)}(X)}{X\mathbb{E}\Big[\frac{1}{X}\Big]}\bigg] \\ &= \bigg(\mathbb{E}\Big[\frac{1}{X}\Big]\bigg)^{-1}\mathbb{E}\bigg[\frac{\chi_{(r,+\infty)}(X)}{X}\bigg] >0. \end{align}

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