I'm looking for convergence results of particular weighted sum: $$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation regarding this kind of sums of random variables?
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$\begingroup$ Try googling for "decoupling inequalities" and/or "U-statistics". In particular, the book of DeLaPena and Gine should be useful to you. $\endgroup$– ofer zeitouniCommented Jul 1, 2015 at 15:44
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1$\begingroup$ It seems like computing the mean and variance of $S_n$ can take you pretty far, and then you can reuse the methods of the law of large numbers to prove convergence with prob 1. Have you tried this? For example, if $E[X^4]$ is finite I think you can get some nice prob 1 convergence results when $a_{ij}$ have some nice properties. $\endgroup$– MichaelCommented Jul 13, 2015 at 21:24
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1 Answer
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Let $A$ be an $n\times n$ matrix with $A_{ij}=a_{ij}/n$. Then $S_n=x^\top A x$, where $x=(X_1,\ldots,X_n)$. In particular, if $A$ is real and symmetric, then you can pass to its spectral decomposition to get
$$S_n=\sum_{i=1}^n\lambda_i\langle x,u_i\rangle^2.$$
This reduces the investigation to understanding the distribution of linear combinations $\langle x,u_i\rangle$, which are perhaps better understood.
answered Jul 1, 2015 at 10:47