A front-loaded Gumbel-like distribution I'm looking for a distribution that is somewhat like the Gumbel distribution and I was wondering if anyone could help.
The parameters are a positive integer $n$ and real numbers $\mu>0$ and $\sigma>0$. Then the distribution is
$$
X \sim \max\Big(\mathcal N(\mu n, \sigma^2n), \mathcal N(\mu(n-1), \sigma^2(n-1)),\ldots, \mathcal N(2\mu, 2\sigma^2), \mathcal N(\mu, \sigma^2)\Big)
$$
where the $\mathcal N$ are independent normal distributions.
Clearly most of the contribution will come from the terms near $\mathcal N(\mu n, \sigma^2n)$, where "near" will depend on the size of $\sigma/\mu$. The parameters I'm looking at right now are around $n=10^7,\ \mu=0.2,\ \sigma=0.4.$
Any information would be good. My particular application requires bounding the tails (finding an upper bound that holds for 1 - 10^-6 or 1 - 10^-9, say) but even help finding basics like the mean and variance would be useful.

The pdf for $X$ can be found but I'm more than skeptical that there is a closed form for the mean and variance.
For example, consider $n=2$.  Using Mathematica one finds the following for the pdf:
n = 2;
dist = OrderDistribution[Table[NormalDistribution[i \[Mu], Sqrt[i] \[Sigma]], {i, n}], n];
PDF[dist, x] /. Erfc[z_] -> 2 \[CapitalPhi][-Sqrt[2] z]

$$\frac{e^{-\frac{(x-2 \mu )^2}{4 \sigma ^2}} \Phi \left(-\frac{\mu -x}{\sigma }\right)}{2 \sqrt{\pi } \sigma }+\frac{e^{-\frac{(\mu -x)^2}{2 \sigma ^2}} \Phi \left(\frac{x-2 \mu }{\sqrt{2} \sigma }\right)}{\sqrt{2 \pi } \sigma }$$
The density contains the standard normal cdf ($\Phi(\dot)$) so you'll likely need to use numerical integration with specific values of $n$, $\mu$, and $\sigma$.
Looking at the pattern of pdf's for several values of $n$, the general form of the pdf appears to be
$$\frac{\left(\prod _{i=1}^n 2 \Phi \left(-\frac{i \mu -x}{\sqrt{i \sigma ^2}}\right)\right) \sum _{i=1}^n \frac{e^{-\frac{(x-i \mu )^2}{2 i \sigma ^2}}}{\sqrt{i} \Phi \left(-\frac{i \mu -x}{\sqrt{i \sigma ^2}}\right)}}{\sqrt{2 \pi } 2^n \sigma }$$
The equation can be put into a more standard form but I'll just leave it with what Mathematica does and avoid messing up the LaTeX.
The cdf has a much simpler form:
$$\prod _{i=1}^n \Phi \left(\frac{x-i \mu}{\sqrt{i} \sigma }\right)$$
