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I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint distribution is unknown but each of them follows the same marginal distribution $F(x)$, I want to prove that an inequality $H(\frac{1}{n-1}\sum_{i=1}^{n-1} E(X_iX_{i+1}))>0$ holds when $n \rightarrow \infty$.

Here is the strategy that I have tried. Supposing $ Y_1, Y_2, \cdots, Y_n $ are individually and identically distributed random variables with distribution function $F(x)$, they can be sorted in an increasing order to form order statistics $Y_{(1)} \leq Y_{(2)} \leq \cdots \leq Y_{(n)}$. Rearranging these order statistics to $Y_{\sigma(1)}, Y_{\sigma(2)}, \cdots, Y_{\sigma(n)}$ by a permutation $\sigma$, I can prove that $$ H(\frac{1}{n-1} \sum_{i=1}^{n-1} E(Y_{\sigma(i)} Y_{\sigma(i+1)})) > 0 $$ holds for any permutation $\sigma$ when $n \rightarrow \infty$.

Now can I use this result to prove that $H>0$ holds for any sequence of random variables with marginal $F(x)$ when $n \rightarrow \infty$ ? If so, could anyone offer some hint on how to do this? I guess copula theory can be used which I am not very familiar with.

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Please tell us what the function $H$ is. – Dilip Sarwate Oct 16 '12 at 13:38
I want to know if this can be generalized regardless of the form of $H$. – RichardKwo Oct 16 '12 at 13:58
If $n=2$, $H$ is a function of one variable. If $n=3$, it is a function of two variables, and in general, $H$ is an arbitrary function of $n-1$ variables. What do you really want to prove? – Dilip Sarwate Oct 16 '12 at 14:00
I see. I re-edited the question. – RichardKwo Oct 16 '12 at 14:07

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