# Convergence of relative sum of iid random variables

I am trying to find some reference with respect to the following problem: Given $$n$$ iid positive random variables $$(X_i)_{i\leq n}$$ of mean $$E(X)>0$$ and variance $$V(X)$$.

I wanted to know if there were some properties on the random variable: $$Z_n = \frac{\sum (X_i -E(X))}{\sum X_i}$$. Specifically I am interested in its convergence speed, something like the law of large numbers or Chebychev/Chernoff bounds. If it does not involve $$V(X)$$ it's even better.

Anyone knows where to look? Thanks

$$Z_n = \frac{\sum_1^n (X_i -E(X))}{\sum_1^n X_i} = 1 - \frac{\mu}{\sum_1^n X_i/n},$$ where $$\mu = E(X)$$.

Then, $$Z = \lim_\limits{n \rightarrow \infty} Z_n =_d 1 - \frac{\mu}{G_n },$$

where the limit is in distribution, with r.v. $$G_n =_d N( \mu, \sigma^2/n)$$

This convergence in distribution implies directly: $$P(|G_n - \mu| >\delta ) = 2\Phi( -\frac{\delta \sqrt{n}}{\sigma})$$

Convergence in distribution to a constant implies convergence in probability (to the same constant), that is, $$G_n \rightarrow_P \mu$$.

Then, $$Z_n$$ converges to zero with rate $$1/n$$ for a fixed error (as can be seen from the simple Wiki proof of weak LLN using Chebyshev inequality).

• Thanks, but this is just a rewrite of the same equation. Can we say something then about the convergence speed of this new RV? – Gopi Mar 23 at 20:50