Let Sn = X1 + ... + Xn denote the nth partial sum of an i.i.d. sequence of random variables having positive mean [mu] and finite variance [sigma]2, and let N(s) = minlcubn [greater-or-equal, slanted] 0: Sn+1 srcub denote the corresponding renewal process. We investigate the strong limiting...

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.

The processes of the form , where K is a constant, and B(·) a Brownian bridge, are investigated. We show that and are both Brownian bridges, and establish the independence of and , this implying that the law of coincides with the conditional law of B, given that . We provide the...

We establish optimal approximation rates of empirical P--P and Q--Q plot processes by sequences of Brownian bridges.

Let Fn and Gn denote the Kaplan-Meier product-limit estimators of lifetime distributions based on two independent samples, and let Fninv and Gninv denote their quantile functions. We consider the corresponding P-P plot Fn(Gninv) and Q-Q plot Fninv(Gn), and establish strong approximations of...