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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Sign up to join this communityI'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?
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.