I have $n$ iid normal random variables $X_i\sim N(0,1)$, and their folded versions $Y_i=|X_i|$. Now I need the cdf, or the density of the sum $$ Y_1+Y_2+\dots+Y_n $$ or at least a good approximation (better than Markov) of $$ \mathbb P (Y_1+Y_2+\dots+Y_n\ge A). $$ I know that for uniform variables there exists the Irwin distribution, so I wonder if there exists an analogue for folded normal variables.
Could anyone help?