Laws of large numbers for bootstrappedU-statistics

A survey of limit laws for bootstrapped sums

“…This unconditional result was also obtained directly, without using the conditional result (3.1), by Athreya et al [13], by Csörgő [29], and by Arenal-Gutiérrez et al [7]. While the proof in [29] is probably the simplest possible, the weak laws in [7] apply to very general parent sequences {X n } of neither necessarily independent nor identically distributed variables.…”

“…holds. On the other hand, Athreya et al [13] and Arenal-Gutiérrez et al [6] obtained conditions under which the unconditional bootstrap SLLN…”

On the strong law for arrays and for the bootstrap mean and variance

“…Athreya, Ghosh, Low and Sen [7] obtained P[sup sup -* kn)_N m)_dn (3.2) as N cx for all > 0 and d > 0 when EIXll a+ < oo for some 6 > 0. Clearly, (3.2) yields the strong consistency of the bootstrap mean ,.,.…”

Maximum likelihood estimation for survey data with informative interval censoring

“…The proof follows the same arguments as that of Theorem 2 but, instead of the classical SLLN, the strong law of large numbers for the bootstrapped mean (see, e.g., Athreya et al 1984) is used.…”