Let $X_1, X_2, ... , X_n$ be i.i.d. normal random variables with mean $0$ and variance $1$. I want to show that the random variables $Y_n := 1/(2n^2) \sum_{i,j =1}^n X_i X_j$ satisfy a large deviation principle with rate function. $I(a)=a$ if $a \geq 0 $ and $I(a) = \infty$ if $ a<0$.
This should be possible by using the contraction principle. However, I do not see how I can use the contraction principle to say something about $X_i*X_j$ and hence about the given sum.