# Is it true that $\mathbb{E}\left[\sum\limits_{k=1}^m \frac{\frac{1}{X_k}}{\sum_{j=1}^m \frac{1}{X_j}}\chi_{(r,+\infty)}(X_k)\right]\to0,m\to\infty?$

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 $$\forall r>0, \mathbb{P}(X0$$.

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

• I removed the "geometric measure theory" tag
– T_M
May 29 '20 at 11:36

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}