Let Sn denote the partial sum of an i.i.d. sequence of centred random variables having a finite moment generating function ø in a neighbourhood of zero. In this paper, we establish strong and weak limit laws for and , where 1[less-than-or-equals, slant]k=k(n)[less-than-or-equals, slant]n is an...

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0,1]2. In particular: (i) we use Fubini-type techniques to...

Let {[alpha]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} and {[beta]n(t),0[less-than-or-equals, slant]t[less-than-or-equals, slant]1} be the empirical and quantile processes generated by the first n observations from an i.i.d. sequence of random variables with a uniform...

We find all possible integral equations among a given family which conserve in distribution the Brownian bridge and the Wiener process. The result implies that the Parzen statistic for goodness of fit can be used if and only if the distribution is exponential, direct or reverse, or uniform.