A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables

We obtain the explicit Karhunen–Loeve decomposition of a Gaussian process generated as the limit of an empirical process based upon independent pairs of exponential random variables. The orthogonal eigenfunctions of the covariance kernel have simple expressions in terms of Jacobi polynomials. Statistical applications, in extreme value and reliability theory, include a Cramér–von Mises test of bivariate independence, whose null distribution and critical values are tabulated.