# pr.probability – $L^p$ inequality for “positively correlated” random variables Answer

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## pr.probability – $L^p$ inequality for “positively correlated” random variables

Suppose that we have $$m$$ complex-valued random variables $$\xi_1,\ldots,\xi_m$$ and assume the following “positive correlation” property: for all non-negative integers $$\alpha_1,\ldots,\alpha_m,\beta_1,\ldots,\beta_m$$ we have
$$\mathbb E(\xi_1^{\alpha_1}\overline{\xi_1^{\beta_1}}\ldots \xi_m^{\alpha_m}\overline{\xi_m^{\beta_m}})\geq 0.$$
Now take any complex numbers $$a_1,\ldots,a_m$$ with $$\max_i|a_i|\leq 1$$ and set
$$M_p(a)=\mathbb E\left|a_1\xi_1+\ldots+a_m\xi_m\right|^p.$$
When $$p=2k$$ other $$k$$ is a positive integer, $$M_p(a)$$ is a polynomial in $$a_i$$ other $$\overline{a_i}$$ with non-negative coefficients, therefore
$$M_{p}(a)\leq M_{p}(1,1,\ldots,1).$$
My question is: will this inequality hold for all real $$p\geq 2$$? And if not, what if we take $$\xi_j=\exp(i\lambda_j N)$$where $$\lambda_j$$ are real and $$N$$ is (the same for different $$j$$) standard normally distributed variable?

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