Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

Hoeffding’s Inequality for Sums of Dependent Random Variables

Let \(X_1,\ldots ,X_n\) be, possibly dependent, [0, 1]-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper, we provide a generalisation of Hoeffding’s theorem. We obtain an estimate on the aforementioned probability that is described in terms of the expectation, with respect to convex functions, of a random variable that concentrates mass on the set \(\{0,1,\ldots ,n\}\). Our main result yields concentration inequalities for several sums of dependent random variables such as sums of martingale difference sequences, sums of k-wise independent random variables, as well as for sums of arbitrary [0, 1]-valued random variables.     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.     

Convergence of weighted sums of tight random elements

Convergence of weighted sums of tight random elements {V_n} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {a_nk} is a Toeplitz sequence of real numbers, then | Σ_k=1^∞a_nkf(V_k)| → 0 in probability for each continuous linear functional f if and only if 6Σ_k=1^∞a_nkV_k 6→ 0 in probability. When the random elements are independent and max_{1≤k≤n | ank | = $\text{O}$(n^?8)} for some $\text{0 <}\text{1}\text{s}\text{< r ? 1}$, then |Σ_k=1^∞a_nkV_k 6→ 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.     

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

A Note on Weighted Sums of Associated Random Variables

We prove the convergence of weighted sums of associated random variables normalized by \({n^{1/p}, p \in}\) (1, 2), assuming the existence of moments somewhat larger than p, depending on the behaviour of the weights, improving on previous results by getting closer to the moment assumption used for the case of constant weights. Besides moment conditions, we assume a convenient behaviour either on truncated covariances or on joint tail probabilities. Our results extend analogous characterizations known for sums of independent or negatively dependent random variables.     

Complete convergence and almost sure convergence of weighted sums of random variables

Letr>1. For eachn1, let {X _nk , –<k<} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee for every >0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.     

Some weak and strong laws of large numbers for D[0,1]-valued random variables

Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D[0,l] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz–Zygmund–Kolmogorov's and Brunk– Chung's Strong Laws of Large Numbers are derived in the setting of D[0,l]-space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology f o r sequences in D[0,l] is studied     

A note on the rates of convergence for weighted sums of ρ * -mixing random variables

We discuss the rates of convergence for weighted sums of ρ ^? -mixing random variables. We solve an open problem posed by Sung [S.H. Sung, On the strong convergence for weighted sums of ρ ^? -mixing random variables, Stat. Pap., 54:773–781, 2013]. In addition, the two obtained lemmas in this paper improve the corresponding ones of Sung in the above-mentioned paper and [S.H. Sung, On the strong convergence for weighted sums of random variables, Stat. Pap., 52:447–454, 2011].     

Weak convergence of products of sums of independent and non-identically distributed random variables

We study limit properties in the sense of weak convergence in the space D[0,1] of certain processes based on products of sums of independent and non-identically distributed random variables. The obtained results extend and generalize results known in the i.i.d. case.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

Laws of large numbers for weighted sums of random variables and random elements

Consider the weighted sums of a sequence {X _n} of independent random variables or random elements inD [0,1]. For convergence ofS _n in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {X _n} is uniformly bounded by a random variableX,i.e.P(¦X _n¦x)P(¦X¦x) for allx>0. Our paper aims at trying to drop this restriction.The Project supported by National Natural Science Foundation of China     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean

The almost sure limiting behavior of weighted sums of independent and identically distributed random variables barely with or without finite mean are established. Results for these partial sums, $$\sum\limits_{k = 1}^n {k^\alpha X_k ,} \alpha \in R$$ have been studied, but only when α=?1 or α=0. As it turns out, the two cases of major interest are α=?1 and α>?1. The purpose of this article is to examine the latter.     

Convergence of series of dependent φ-subgaussian random variables

The almost sure convergence of weighted sums of φ-subgaussian m-acceptable random variables is investigated. As corollaries, the main results are applied to the case of negatively dependent and m-dependent subgaussian random variables. Finally, an application to random Fourier series is presented.     

Self-Normalized LIL for Hanson–Russo Type Increments

Let X ₁, X ₂,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given.     

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f ^?1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.     

A central limit theorem for D(A)-valued processes

Let D(A) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D(A) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.     

Algorithmic Chernoff-Hoeffding inequalities in integer programming

Proofs of classical Chernoff-Hoeffding bounds has been used to obtain polynomial-time implementations of Spencer's derandomization method of conditional probabilities on usual finite machine models: given m events whose complements are large deviations corresponding to weighted sums of n mutually independent Bernoulli trials. Raghavan's lattice approximation algorithm constructs for 0-1 weights, and integer deviation terms in O(mn)-time a point for which all events hold. For rational weighted sums of Bernoulli trials the lattice approximation algorithm or Spencer's hyperbolic cosine algorithm are deterministic precedures, but a polynomial-time implementation was not known. We resolve this problem with an O(mn²log $ {{mn}\over{\epsilon}} $)-time algorithm, whenever the probability that all events hold is at least ϵ > 0. Since such algorithms simulate the proof of the underlying large deviation inequality in a constructive way, we call it the algorithmic version of the inequality. Applications to general packing integer programs and resource constrained scheduling result in tight and polynomial-time approximations algorithms. © 1996 John Wiley & Sons, Inc.     

L 1 bounds for asymptotic normality of random sums of independent random variables

We present upper bounds of L ₁ norms in the central limit theorem for random sums of independent random variables X ₁ , X ₂ , . . . , where the number of summands N is a nonnegative integer-valued random variable, independent of the summands X ₁ , X ₂ , . . . .