What is the pdf of $N(0,\sigma^2)+\max\{N(0,\sigma^2),...,N(0,\sigma^2)\}$? Define the following random variables
1) $\eta\sim N(0,1)$
2) $\zeta\sim N(0,1)$
3) $\tau_k\sim N(0,1)$ for $k=1,...,K$
4) $\epsilon_k\sim N(0,1)$ for $k=1,...,K$
5) $\tau\equiv \max\{\tau_1,..., \tau_K\}$
6) $\epsilon \equiv \max\{\epsilon_1,..., \epsilon_K\}$
We also assume that $(\eta, \zeta, \tau_1,...,\tau_K, \epsilon_1,..., \epsilon_K)$ are mutually independent.
Question: Can we say something about the probability distribution of
$$
-\eta-\zeta+\tau+\epsilon
$$
?
I know that $-\eta-\zeta\sim N(0,2)$. My doubts are about the distribution of $\eta+\tau$ and what happens when we add that to a normal. 

Use $$\mathbb{P}[\max(X_1,\ldots,X_m)\leq a]=\prod_{i=1}^m \mathbb{P}[X_i\leq a]$$
which holds for any collection of mutually independent random variables and the symmetry of zero mean Gaussians to proceed.
Distribution of sums is the convolution of distributions.
