Past exam of the random matrix theory course of the University of Oxford / 2025 / 1 / a / ii / Solution

The joint probability distribution of all entries of $X$ is unchanged by $X\mapsto -X$, so $M$ and $-M$ have the same probability distribution. For the specified cyclic product $P=M_{i_1{i}_2}\cdots M_{i_k{i}_1}$, this transformation multiplies $P$ by $(-1{)}^k$. Every such product is an integrable random variable, since normal random variables have finite moments of all orders. Thus, for odd $k$,Repeated indices cause no difficulty: all factors change sign simultaneously. It would be incorrect to assume that these factors are independent.