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 $q_n:=\dfrac{1}{n}\sum_{i=1}^n\sum_{j=1}^nh_{i,j}x_ix_j$.

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 ?

**Note.** In the special case $h_{ij} = \delta_{ij}$, we have $q_n = \dfrac{1}{n}\sum_{i=1}^n x_i^2 \overset{p}{\longrightarrow}1$.