Convergence rate of the dependent bootstrapped means

In this paper, a Baum–Katz, Erdos, Hsu–Robbins, Spitzer type complete convergence result is obtained for the dependent bootstrapped means.National Sciences and Engineering Research Council of Canad

Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces

For an array of rowwise independent random elements {Vnj , j ≥ 1, n ≥ 1} in a real separable, stable type p Banach space X and an array of constants {anj , j ≥ 1, n ≥ 1}, general weak laws of large numbers of the forms (i) Pkn j=1 anjVnj P→ 0 and (ii) PTn j=1 anj (Vnj −cnj ) P→ 0 are obtained where for (i), EVnj = 0, j ≥ 1, n ≥ 1 and the kn are permitted to assume the value ∞ and for (ii), {cnj , j ≥ 1, n ≥ 1} is a suitable array of elements in X and {Tn, n ≥ 1} is a sequence of positive integer-valued random variables (called random indices). In the main results, the random elements {Vnj , j ≥ 1, n ≥ 1} are assumed to be stochastically dominated by a random element V and the hypotheses impose conditions on the growth behavior of the {anj , j ≥ 1, n ≥ 1}, on the tail of the distribution of ||V ||, and (for (ii)) on the marginal distributions of the random indices. The results of the form (i) are shown to be valid for a mode of convergence which is stronger than convergence in probability, viz. convergence in the Lorentz space L(p,∞)(X ). It is shown via example that the stable type p hypothesis cannot be relaxed