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I came across a problem involving the following limit:

$\lim_{n \to \infty} (\frac{1}{n} \sum\limits_{i=1}^n \mathbf{1}_{x_i>0}), \mbox{ where } X \sim N(\mu, \sigma)$

How would you approach evaluating the limit of this sum? I thought about applying some form of Riemann integral, but got stuck with the indicator function... Also, is it possible to say something about the distribution of the sum?

Thanks a lot!

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Can you interpret this limit as the probability of some event? (Hint: yes.) – Michael Lugo Aug 25 '12 at 15:01
This is actually part of the background to my question - the above is part of the broader problem. I've got an event which is determined by the sum $y_i \equiv k - c_i - \phi l$ being positive, where $C \sim N(\mu, \sigma)$ and k and l being constant. I want to express the probability of this event occuring depending on $\phi$, which shifts the mean of the distribution of $Y$. From this, can I simply claim that $Y \sim N(\mu + k - \phi l, \sigma)$? As a more general question, would it make sense at all to look for a derivative of something like $Prob(Y)*m$, m being some constant? Thanks a lot! – jush Aug 25 '12 at 15:55
if the $X_i$ are i.i.d, as I suppose the are, the summands are i.i.d random variables. use the SLLN – mike Aug 25 '12 at 19:32
up vote 0 down vote accepted

One is considering $T_n=\frac{1}{n} \sum\limits_{i=1}^nY_i$, with $Y_i= \mathbf{1}_{X_i>0}$, for some random variables $(X_i)_i$. If the random variables $(X_i)_i$ are i.i.d., the random variables $(Y_i)_i$ are, and the strong law of large numbers shows that $T_n\to\mathrm E(Y)=\mathrm P(X\gt0)$, almost surely and in every $L^p$ for $p$ finite.

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