Let $X_1,X_2,...,X_n$ be iid standard Gaussian random variables.
Consider the set of random variables $M =\left\{\left( X_i-X_j\right) :i,j = \left\{1,2,\dots,n\right\} \& i\ne j\right\}$.
I am trying to find a closed form expression for the cdf or an analytical expression on the upper bound of the cdf of the maximum of $M$. How do I take care of the correlation between the variables in $M$? Is there any asymptotic results as $n \rightarrow \infty$ ?
Hint: If you are interested only in distribution of maximum of the differences than notice $$ \max_{i,j=1,\ldots,n ~ \& ~i\ne j} \left\{\left( X_i-X_j\right) \right\} = \max_{i=1,\ldots,n}\left\{X_i\right\} - \min_{{i=1,\ldots,n}}\left\{X_i\right\}. $$ So all you have to do is find a joint distribution of minimal and maximal value of $X_1,\ldots,X_n$.