Let $(h\_{ij})\_{i,j \in \mathbb N}$ be a sequence of real numbers (deterministic) and let $x\_1,\ldots,x\_n,\ldots$ be a sequence of iid $N(0,1)$ randm variables. For each positive integer $n$, consider the quadratic form $q\_n:=\dfrac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^nh\_{i,j}x\_ix\_j$.



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> **Question.** Under what conditions on the sequence $(h\_{ij})$ does there exist $c \ge 0$ sucht aht $q\_n \to c$ in probability ? Is there some other kind of convergence that might hold here ?
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**Note.** In the special case $h\_{ij} = \delta\_{ij}$, we have $q\_n = \dfrac{1}{n}\sum\_{i=1}^n x\_i^2 \overset{p}{\longrightarrow}1$.



When does a gaussian quadratic form converge (in probability) to a constant?
First, notice that w.l.o.g. you can assume that the matrix $H\_n$ is diagonal (from rotational invariance of the isotropic Gaussian).


You thus are interested in
$$
\frac{1}{n} \sum\_{i=1}^n \lambda\_i(H\_n) X\_i^2.
$$


So the condition
$$\sum\_{i=1}^n \lambda\_i^2(H\_n)/n^2 \to 0$$


is the key.


Then, one has that
$$
Var\left(\frac{1}{n} \sum\_{i=1}^n \lambda\_i(H\_n) X\_i^2\right)= 2\sum\_{i=1}^n \lambda\_i^2(H\_n)/n^2 \to 0. 
$$
and
$$
\mathbb{E}\left(\frac{1}{n} \sum\_{i=1}^n \lambda\_i(H\_n) X\_i^2\right)= \sum\_{i=1}^n \lambda\_i(H\_n)/n. 
$$


