On the asymptotic behaviour of the joint distribution of the maxima and minima of observations, when the sample size is a random variable (Q6618822)

Let \(X_1,X_2,\ldots\) be a sequence of independent and identically distributed random variables, and \(N_1,N_2,\ldots\) be a sequence of positive integer-valued random variables independent of the \(X_j\). With \(m\) and \(M\) denoting the minimum and maximum, respectively, of \(X_1,\ldots,X_{N_n}\) the author studies the limiting joint distribution of suitably normalised versions of \(m\) and \(M\), and the limiting distribution of their difference, as \(n\to\infty\). An explicit expression is given for the joint distribution of \((M,m)\) when \(N_n/n\) converges in distribution as \(n\to\infty\), and it is further shown that the limiting random variables are independent if and only if \(N_n/n\) converges in probability to a positive constant. Under this latter assumption, the limiting distribution of the range \(M-m\) is also derived. Several examples are considered throughout the paper.