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Is there any chance to tackle something like

$$ \mathbb{E}\left( \left| \sum_{i=1}^N X^{p_i} Y^{q_i} \right| \right) $$

where $X$ and $Y$ are standard Gaussian variables?

Note that I am explicitely adding the powers and two random variables (I would love to be able to bring the $\mathbb{E}$ somehow inside the sum). Without it, the solution seems simple: The expectation of absolute value of random variables

I am also aware that I could apply the triangle inequality to get an upper bound. However, this is too unprecise for me.

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  • $\begingroup$ Do you assume $X$ and $Y$ independent? Also, note that when all $p_i,q_i$ are even integers, the absolute values do not matter and your problem reduces to finding $\mathbb{E}[X^p]$ for which formulas exist. $\endgroup$
    – Raskolnikov
    Commented Jul 9, 2019 at 14:00
  • $\begingroup$ Yes $X$ and $Y$ are independent. Unfortunately $p_i$ and $q_i$ are even AND odd (but positive integers) $\endgroup$
    – divB
    Commented Jul 9, 2019 at 17:51
  • $\begingroup$ @Raskolnikov Why can we disregard the absolute value when $p_i$ and $q_i$ are integers? $\endgroup$
    – sonicboom
    Commented Mar 31, 2021 at 10:35
  • $\begingroup$ @sonicboom: I said EVEN integers. $\endgroup$
    – Raskolnikov
    Commented Mar 31, 2021 at 12:34

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