Motivated by our earlier work on change-point analysis we prove a number of limit theorems for increments of renewal counting processes, or the corresponding first passage times. The starting point of the increments is deterministic as well as random, a typical example being the first stopping...

We obtain a lower bound for the rate of approximation of bootstrapped empirical processes with Brownian bridges. Our result corresponds to the optimality of the Komlós et al. (1975, 1976, Z. Wahrsch. Verw. Gebiete 32, 111-131; 34, 33-58) approximation of empirical processes.

Let {(Xi,Yi)}i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y1=y)=0 for all y. Put Mn=Mn(Ln)=max0[less-than-or-equals, slant]k[less-than-or-equals, slant]n-Ln(Xk+1+...+Xk+Ln)Ik,Ln, where Ik,l=I{Yk+1[less-than-or-equals, slant]...[less-than-or-equals, slant]Yk+l} denotes the...

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...