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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

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

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

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