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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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    if the $X_i$ are i.i.d, as I suppose the are, the summands are i.i.d random variables. use the SLLN2012-08-25

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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.