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!