Looking for $\mathrm{E}[\min(\sum{X}) ]$ and $\mathrm{E}[\max(\sum{X})]$. Paper references much appreciated.
Model: let's say we have 3 connected devices in a signal processing pipeline:
$$ \underset{\mu,\sigma,\min,\max}{\text{sampler}} \rightarrow (X)\rightarrow \underset{\text{over }k}{\text{summator}} \rightarrow(Y)\rightarrow \underset{\text{over }m}{\text{extremer}} \rightarrow(Z^\min,Z^\max)\rightarrow $$
1) sampler is producing random signal series $X_{t1}$ with period $T_X=1$, and $X$ has stable distribution, with known mean $\mu$, variance $\sigma^2$, minimum $\min$, maximum $\max$.
2) summator is looking at $X$ and producing series $Y_{t2}=\sum{X}_{n}$ with period $T_Y=k*T_X=k$, $k>1$, creating new sum over every bucket of $k$ samples of $X$.
3) extremer is looking for $Y$ boundaries and producing 2 series $Z^\min_{t3}=\min(Y_n)$ and $Z^\max_{t3}=\max(Y_n)$, with period $T_Z=m*T_Y=m*k$, $m>1$, creating new minimum and maximum over every bucket of $m$ samples of $Y$.
How one obtains expectations $E_\min=\mathrm{E}[Z^\min]$ and $E_\max=\mathrm{E}[Z^\max]$ based on known parameters $X$ $(\mu,\sigma,\min,\max)$ and frequency parameters $(k,m)$?
NOTE: Current answer does not take into account $\min$ and $\max$ assumption. Probably an approach similar to Irwin–Hall distribution calculation would give a better estimate. What do you think?
It looks pretty hard to derive a closed expression for this. What I might do if I had to have an answer is assume k is big, apply the Central Limit Theorem to the sum, and then use the approximation discussed here to yank an order statistic from it: https://stats.stackexchange.com/questions/9001/approximate-order-statistics-for-normal-random-variables
Some references are cited in that discussion