Suppose $X_1, \dots, X_n$ are truncated standard normal variables, truncated so that $X_i \geq 0$ (that is, $X_i$ is drawn as a standard normal, conditional on $X_i \geq 0$)
Let $c_1, \dots, c_n$ be non-negative coefficients.
What does the distribution of $\sum_i c_i Y_i$ look like? Does it have, or approximately have, a standard distribution, such as a truncated normal distribution?
Original question:
Suppose $X_1, \dots, X_n$ are iid Normal random variables, with mean 0 and variances $\sigma_1, \dots, \sigma_n$.
Let $Y_i = \max(0,X_i)$. (So $Y_i$ is a truncated normal random variable).
What does the distribution of $\sum_i Y_i$ look like? Does it have, or approximately have, a standard distribution?