Andrew Rosalsky

On complete convergence for arrays of rowwise independent random elements in Banach spaces

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established

Strong laws of large numbers for weighted sums of random elements in normed linear spaces.

Consider a sequence of independent random elements {Vn, n > in a real separable normed linear space (assumed to be a Banach space in most of the results), and sequences of con-

Some general strong laws for weighted sums of stochastically dominated random variables

For weighted sums, general strong laws of large numbers of the form almost certainly are established where the random variables are stochastically dominated by a random variable are suitable conditional expectations or are all 0. The hypotheses involve both the behavior of the tail of the distribution of ∣Y∣ and the growth behavior of the constants bn/an. In general, no assumptions are made regarding the joint distributions of . As special cases, both old and new results are obtained

On the rate of complete convergence for weighted sums of arrays of banach space valued random elements

By applying a recent result of Hu et al. [Stochastic Anal. Appl., 17 (1999), pp. 963--992], we extend and generalize the complete convergence results of Pruitt [ J. Math. Mech., 15 (1966), pp. 769--776] and Rohatgi [Proc. Cambridge Philos. Soc., 69 (1971), pp. 305--307] to arrays of row-wise independent Banach space valued random elements. No assumptions are made concerning the geometry of the underlying Banach space. Illustrative examples are provided comparing the various results.

A survey of limit laws for bootstrapped sums

Concentrating mainly on independent and identically distributed (i.i.d.) real-valued parent sequences, we give an overview of first-order limit theorems available for bootstrapped sample sums for Efron’s bootstrap. As a light unifying theme, we expose by elementary means the relationship between corresponding conditional and unconditional bootstrap limit laws. Some open problems are also posed.

A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces

For weighted sums of the form Sn = [summation operator]j=1kn anj(Vnj - cnj) where {anj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]kn

A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces

For weighted sums [Sigma]j = 1najVj of independent random elements {Vn, n >= 1} in real separable, Rademacher type p (1 p 0 is established, where {vn, n >= 1} and bn --> [infinity] are suitable sequences. It is assumed that {Vn, n >= 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of V and the growth behaviors of the constants {an, n >= 1} and {bn, n >= 1}. No assumption is made concerning the existence of expected values or absolute moments of the {Vn, n >- 1}.

Strong and Weak Laws of Large Numbers for Double Sums of Independent Random Elements in Rademacher Type p Banach Spaces

Abstract For a double array of independent random elements {V mn , m ≥ 1, n ≥ 1} in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space, strong and weak laws of large numbers are established for the double sums , m ≥ 1, n ≥ 1. For the strong law result, it is assumed that EV mn = 0, m ≥ 1, n ≥ 1 and conditions are provided under which almost surely as max {m, n} → ∞. These conditions for the strong law are shown to completely characterize Rademacher type p Banach spaces. The weak law results provide conditions for as max {m, n} → ∞ or for as max {m, n} → ∞ to hold where c ijmn = E(V ij I(‖V ij ‖ ≤ mn)), i, j, m, n ≥ 1 and {T m , m ≥ 1} and {τ n , n ≥ 1} are sequences of positive integer-valued random variables. Illustrative examples are provided.

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {Wnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[less-than-or-equals, slant]1} and {Xnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[greater-or-equal, slanted]1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.

On the weak law of large numbers for normed weighted sums of I.I.D. random variables

For weighted sums ∑j=1najYj of independent and identically distributed random variables {Yn,n≥1}, a general weak law of large numbers of the form (∑j=1najYj−νn)/bn→P0 is established where {νn,n≥1} and {bn,n≥1} are statable constants. The hypotheses involve both the behavior of the tail of the distribution of |Y1| and the growth behaviors of the constants {an,n≥1} and {bn,n≥1}. Moreover, a weak law is proved for weighted sums ∑j=1najYj indexed by random variables {Tn,n≥1}. An example is presented wherein the weak law holds but the strong law fails thereby generalizing a classical example.