A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, 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 {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

Divergence Criterion for a Class of Random Series Related to the Partial Sums of I.I.D. Random Variables

Journal of Theoretical Probability - Let $$ \{X, X_{n};~n \ge 1 \}$$ be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to...     

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 {W_nj, 1jn, n1} and {X_nj, 1jn, n1} 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.     

A generalization of the Iterated Logarithm Law for weighted sums with infinite variance

Summary A generalization of the classical Law of the Iterated Logarithm (LIL) is obtained for the weighted i.i.d. case consisting of sequences { _n Y _n } where the weights { _n } are nonzero constants and {Y _n} are i.i.d. random variables. If Y is symmetric but not necessarily square integrable and if the weights satisfy a certain growth rate, conditions are given which guarantee that { _n Y _n} obey a Generalized Law of the Iterated Logarithm (GLIL) in the sense that almost certainly for some positive conslants a _n . Teicher has shown that such weights entail the classical LIL when EY ²< and Feller has treated the GLIL when _n =1 and EY ²=. The main finding here asserts that if {q_n} satisfies q _n ² =nG(q_n)loglogq _n where G is a specified slowly varying function, asymptotically equivalent to the truncated second moment of Y, and if a certain series converges, then the GLIL obtains with where .     

Aggregation and the estimation of equation systems

Simulations based on two sets of data are used to assess the impact of combining equations on the accuracy of parameter estimates and their asymptotic standard errors.     

The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array of i.i.d. random variables with mean and variance and set . Let denote the empirical distribution function of Z_{1, n} ,..., Z _{N, n} and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for , where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.