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

Abstract

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.