In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than ...that obtained for the real-valued random variables by Chen et al.
, J. Math. Anal. Appl.
(2014), 1290–1302, and Chen and Sung
, J. Math. Inequal.
(3) (2016), 837–848. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.
The classical Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of n mutually independent and identically ...distributed random variables with finite second moment converges in distribution to a standard Gaussian as n goes to infinity. In particular, pairwise independence of the sequence is generally not sufficient for the theorem to hold. We construct explicitly such a sequence of pairwise independent random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions) and for which no CLT holds. We obtain, in closed form, the asymptotic distribution of the sample mean of our sequence, and find it is asymmetrical for any F. This is illustrated through several theoretical examples for various choices of F. Associated R codes are provided in a supplementary appendix online.
A number of standard textbooks that are followed in a junior/senior level course or in a first-year graduate level course in mathematical statistics and probability, routinely include one single ...basic illustration, obviously in its variant forms, to highlight an important point: pairwise independence may not imply (mutual) independence. We earnestly believe that beginning students appreciate more examples to clarify these key issues. Hence, we hope that our new sets of nontrivial illustrations from Section 2 will help our audience. Next, in Section 3, we extend the notion to q-wise independence with a large set of illustrations using both discrete and continuous random variables showing that q-wise independence may not imply (mutual) independence. We reasonably assure that this discourse is immediately accessible to juniors/seniors and first-year graduate students.
The authors develop a framework to select a subset of sensors from a field in which the sensors have an ingrained independence structure. Given an arbitrary independence pattern, the authors ...construct a graph that denotes pairwise independence between sensors, which means those sensors may operate simultaneously without interfering. The set of all fully-connected subgraphs (cliques) of this independence graph forms the independent sets of matroids over which the authors maximise the average and minimum of a set of submodular objective functions. The average case is submodular, so it can be approximated. The minimum case is both non-submodular and inapproximable. The authors propose a novel algorithm GENSAT that exploits submodularity and, as a result, returns a near-optimal solution with approximation guarantees on a relaxed problem that are within a small factor of the average case scenario. The authors apply this framework to ping sequence optimisation for active multistatic sonar arrays by maximising sensor coverage for average and minimum case scenarios and derive lower bounds for minimum probability of detection for a fractional number of targets. In these ping sequence optimisation simulations, GENSAT exceeds the fractional lower bounds and reaches near-optimal performance, and submodular function optimisation vastly outperforms traditional approaches and nearly achieves optimal performance.
Estimation of Distribution Algorithms (EDAs) were proposed as an alternative for traditional evolutionary algorithms in which reproduction operators could rely on information extracted from the ...population to enable a more effective search. Since information is usually represented as a probabilistic graphic model, the effectiveness of EDAs strongly depends on how accurately such models represent the population. In this sense, models of increasing complexity have been employed by EDAs, with the most successful ones being able to encode multivariate factorizations of joint probability distributions. However, some studies have shown that even multivariate EDAs fail to build accurate models for problems in which there is an intrinsic pairwise independence between variables. This study elucidates how pairwise independence impacts the linkage learning procedures of multivariate EDAs and affects their accuracy. First, the necessary conditions for learning additively separable functions are assessed, from which it is shown that extreme multimodality can induce pairwise independence. Second, it is demonstrated that in the presence of pairwise independence the approximate linkage learning procedures employed by many EDAs are not able to retrieve high-order dependences. Finally, in an attempt to infer how likely pairwise independence occur in practical problems, the case of non-separable functions is empirically investigated. For this purpose, the NK-model and the Linkage-Tree Genetic Algorithm (LTGA) were used as a study case and a range of usefulness for the LTGA was estimated according to N (problem size) and K (degree of interactions among variables and multimodality). The results indicated that LTGA linkage learning is probably more useful for K≤6 on instances with random linkages (this range grows with N), and for K≤9 on instances with nearest-neighbor linkages (this range is stable with N). Outside these ranges, pairwise independence is more likely to occur, which deteriorates models accuracy and impairs LTGA performance.
Using the technique of finding bounds on sets of copulas with particular properties, we compare the distribution of an n-dimensional (n≥3) vector of continuous pairwise independent random variables ...to the distribution of a similar vector of mutually independent random variables. We examine the n=3 case in detail, and provide asymptotic results in the general case.
Generalizations of the second Borel–Cantelli lemma are obtained under very weak dependence conditions, subsuming several earlier results as special cases.
We show that for a large class of positive weights including the ones that are eventually monotone decreasing and those that are eventually monotone increasing but vary regularly, if the averages of ...random variables converge in some sense, then their corresponding weighted averages also converge in the same sense. We will also replace the sufficient conditions in the fundamental result of Jamison, Pruitt, and Orey for i.i.d. random variables that make their work more transparent.
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