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Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \ldots \oplus c_k}$, the joint moment $\mathrm{E}(f_{c_1}\cdot\ldots\cdot f_{c_k})$ of $k$ such characters is 1 if $c_1 \oplus \ldots \oplus c_k = 0$, and 0 otherwise, when the expectation is taken over the uniform distribution.

Is there a similar simple characterization of the joint cumulant $\kappa(f_{c_1},\ldots,f_{c_k})$?

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  • $\begingroup$ The product of any k characters is again a character. How do you get anything from joint moments beyond what happens for single characters? $\endgroup$ Oct 14, 2015 at 18:37
  • $\begingroup$ I don't, this is exactly why it's simple in this case - it reduces to expectation over a single character. $\endgroup$
    – R S
    Oct 15, 2015 at 21:19

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