Pairwise Independent Research Articles

The essential equivalence of pairwise and mutual conditional independence

For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent — i.e., equivalent to up to null sets. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our result relies on an iterated extension of Bledsoe and Morse's completion of the product of two measure spaces.

The exact law of large numbers via Fubini extension and characterization of insurable risks

A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product.

Strong Laws of Large Numbers by Elementary Tauberian Arguments

For Kolmogorovs strong law of large numbers an alternative short proof is given which weakens Etemadis condition of pairwise independence. The argument uses the known ? and elementary ? equivalence of (Cesaro) C1- and C2-summability for one-sided bounded sequences. Also other strong laws of large numbers are established, in part via Borel summability.

G-monotonicity and G-convexity

The concept of monotonicity and its various extensions existing in the literature in the classical setting are generalized in finite dimensional Euclidean spaces. With examples, as support to the proposed structure, it is shown when the traditional development fails but the proposed one works. As an application, the computation of probability under pairwise disjointness and mutual independence for a finite number of sets is shown to be tied to the proposed framework. Characterizations are given in terms of generalized convexity and its extension. Some basic results are shown which may point to applications to some of the popular areas of research in theoretical mathematical programming

A robust image fingerprinting system using the Radon transform

With the ever-increasing use of multimedia contents through electronic commerce and on-line services, the problems associated with the protection of intellectual property, management of large database and indexation of content are becoming more prominent. Watermarking has been considered as efficient means to these problems. Although watermarking is a powerful tool, there are some issues with the use of it, such as the modification of the content and its security. With respect to this, identifying content itself based on its own features rather than watermarking can be an alternative solution to these problems. The aim of fingerprinting is to provide fast and reliable methods for content identification. In this paper, we present a new approach for image fingerprinting using the Radon transform to make the fingerprint robust against affine transformations. Since it is quite easy with modern computers to apply affine transformations to audio, image and video, there is an obvious necessity for affine transformation resilient fingerprinting. Experimental results show that the proposed fingerprints are highly robust against most signal processing transformations. Besides robustness, we also address other issues such as pairwise independence, database search efficiency and key dependence of the proposed method.

A derandomization using min-wise independent permutations

Min-wise independence is a recently introduced notion of limited independence, similar in spirit to pairwise independence. The latter has proven essential for the derandomization of many algorithms. Here we show that approximate min-wise independence allows similar uses, by presenting a derandomization of the RNC algorithm for approximate set cover due to S. Rajagopalan and V. Vazirani. We also discuss how to derandomize their set multi-cover and multi-set multi-cover algorithms in restricted cases. The multi-cover case leads us to discuss the concept of k-minima-wise independence, a natural counterpart to k-wise independence.

INDEPENDENCE OF DOUBLE WIENER INTEGRALS

In this paper a necessary and sufficient condition is obtained for two double Wiener integrals to be statistically independent, first in the case of symmetric and continuous kernels. It is also shown that, for more than two double Wiener integrals, pairwise independence implies mutual independence. After that, the continuity condition on the kernels is somewhat relaxed, and it is shown that Craig's (1943, Annals of Mathematical Statistics 14, 195–197) theorem on the independence of quadratic forms in normal variables is a special case of the result obtained for the case of discontinuous kernels.

Averaging probability judgments: Monte Carlo analyses of asymptotic diagnostic value

Wallsten et al. (1997) developed a general framework for assessing the quality of aggregated probability judgments. Within this framework they presented a theorem regarding the effects of pooling multiple probability judgments regarding unique binary events. The theorem states that under reasonable conditions, and assuming conditional pairwise independence of the judgments, the average probability estimate is asymptotically perfectly diagnostic of the true event state as the number of estimates pooled goes to infinity. The purpose of the present study was to examine by simulation (1) the rate of convergence of averaged judgments to perfect diagnostic value under various conditions and (2) the robustness of the theorem to violations of its assumption that the covert probability judgments are conditionally pairwise independent. The results suggest that while the rate of convergence is sensitive to violations of the conditional pairwise independence, the asymptotic properties remain relatively robust under a large variety of conditions. The practical implications of these results are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Averaging probability judgments: Monte Carlo analyses of asymptotic diagnostic value

AbstractWallsten et al. (1997) developed a general framework for assessing the quality of aggregated probability judgments. Within this framework they presented a theorem regarding the effects of pooling multiple probability judgments regarding unique binary events. The theorem states that under reasonable conditions, and assuming conditional pairwise independence of the judgments, the average probability estimate is asymptotically perfectly diagnostic of the true event state as the number of estimates pooled goes to infinity. The purpose of the present study was to examine by simulation (1) the rate of convergence of averaged judgments to perfect diagnostic value under various conditions and (2) the robustness of the theorem to violations of its assumption that the covert probability judgments are conditionally pairwise independent. The results suggest that while the rate of convergence is sensitive to violations of the conditional pairwise independence, the asymptotic properties remain relatively robust under a large variety of conditions. The practical implications of these results are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Maine Caucasian Population DNA Database Using Twelve Short Tandem Repeat Loci

A population study of Caucasians residing in Maine was conducted using the AmpF1STR Profiler PCR Amplification Kit and the AmpF1STR Profiler Plus PCR Amplification Kit (Applied Biosystems Division (ABD) of Perkin Elmer, Foster City, CA). The kits contain the reagents necessary to amplify 12 different STR loci and the gender marker Amelogenin using two multiplex PCR, each containing nine STR loci. Thus, there is an overlap of six STR loci. The 12 STR loci are TH01, TPOX, CSF1PO, D3S1358, vWA, FGA, D8S1179, D21S11, D18S51, D5S818, D13S317, and D7S820. These loci represent 12 of the 13 core loci selected by the CODIS STR standardization project. Dye-labeled amplification products were separated and detected using the capillary electrophoresis instrument ABI Prism 310 Genetic Analyzer. Allele frequencies were determined for the 12 STR loci. Statistical analysis of the data included Hardy-Weinburg equilibrium (HWE) analysis, pairwise independence testing, power of discrimination (PD), and probability of exclusion (PE).

The almost equivalence of pairwise and mutual independence and the duality with exchangeability

For a large collection of random variables in an ideal setting, pairwise independence is shown to be almost equivalent to mutual independence. An asymptotic interpretation of this fact shows the equivalence of asymptotic pairwise independence and asymptotic mutual independence for a triangular array (or a sequence) of random variables. Similar equivalence is also presented for uncorrelatedness and orthogonality as well as for the constancy of joint moment functions and exchangeability. General unification of multiplicative properties for random variables are obtained. The duality between independence and exchangeability is established through the random variables and sample functions in a process. Implications in other areas are also discussed, which include a justification for the use of mutually independent random variables derived from sequential draws where the underlying population only satisfies a version of weak dependence. Macroscopic stability of some mass phenomena in economics is also characterized via almost mutual independence. It is also pointed out that the unit interval can be used to index random variables in the ideal setting, provided that it is endowed together with some sample space a suitable larger measure structure.

Fast Distributed Algorithms for Brooks-Vizing Colourings (Extended Abstract)

Fast Distributed Algorithms for Brooks-Vizing Colourings (Extended Abstract)

An $\NC$ Algorithm for Minimum Cuts

We show that the minimum-cut problem for weighted undirected graphs can be solved in $\NC$ using three separate and independently interesting results. The first is an $(m^2/n)$-processor $\NC$ algorithm for finding a $(2+\epsilon)$-approximation to the minimum cut. The second is a randomized reduction from the minimum-cut problem to the problem of obtaining a $(2+\epsilon)$-approximation to the minimum cut. This reduction involves a natural combinatorial set-isolation problem that can be solved easily in $\RNC$. The third result is a derandomization of this $\RNC$ solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the set-isolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe $\NC$ algorithms finding minimum $k$-way cuts for any constant $k$ and finding all cuts of value within any constant factor of the minimum. Another application of these techniques yields an $\NC$ algorithm for finding a {\em sparse $k$-connectivity certificate} for all polynomially bounded values of $k$. Previously, an $\NC$ construction was only known for polylogarithmic values of $k$.

Hyperfinite Law of Large Numbers

AbstractThe Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. An asymptotic interpretation of results established here shows that for a triangular array (or a sequence) of random variables, asymptotic uncorrelatedness or asymptotic pairwise independence is necessary and sufficient for the validity of appropriate versions of the law of large numbers. Our intrinsic characterization of almost sure pairwise independence leads to the equivalence of various multiplicative properties of random variables.

On the impossibility of amplifying the independence of random variables

AbstractIn this paper we prove improved lower and upper bounds on the size of sample spaces which which are required to be independent on specified neighborhoods. Our new constructions yield sample spaces whose size is smaller than previous constructions due to Schulman. Our lower bounds generalize the known lower bounds of Alon et al. and Chor et al. In obtaining these bounds we examine the possibilities and limitations of amplifying limited independence by fixed functions. We show that in general independence cannot be amplified from k‐wise independence to (k + 1)‐wise independence. Finally, we enumerate all possible logical consequences of pairwise independence random bits, i.e., events whose probabilities are a consequence of pairwise independence.

The Role of Geometry in Pairwise and Mutual Independence

Abstract The concepts of pairwise and mutual independence, while fundamental to probability theory, are sometimes difficult for students to differentiate. For pairwise independent random variables uniformly distributed over their region of support, a “rectangular support”—a Cartesian product of intervals—for the joint density is a necessary and sufficient condition for mutual independence. The use of geometric illustrations will help students visualize this result and provide new insight into the difference between these two concepts.

The Role of Geometry in Pairwise and Mutual Independence

The Role of Geometry in Pairwise and Mutual Independence

Removing randomness in parallel computation without a processor penalty

We develop some general techniques for converting randomized parallel algorithms into deterministic parallel algorithms without a blowup in the number of processors. One of the requirements for the application of these techniques is that the analysis of the randomized algorithm uses only pairwise independence. Our main new result is a parallel algorithm for coloring the vertices of an undirected graph using at most Δ + 1 distinct colors in such a way that no two adjacent vertices receive the same color, where Δ is the maximum degree of any vertex in the graph. The running time of the algorithm is O(log 3 n log log n) using a linear number of processors on a concurrent read, exclusive write (CREW) parallel random access machine (PRAM). Our techniques also apply to several other problems, including the maximal independent set problem and the maximal matching problem. The application of the general technique to these last two problems is mostly of academic interest because parallel algorithms that use a linear number of processors which have better running times have been found previously.

A stationary, pairwise independent, absolutely regular sequence for which the central limit theorem fails

A strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. The mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of Ibragimov.

A De Finetti Theorem for a Class of Pairwise Independent Stationary Processes

Consider a $\{0, 1\}$-valued strictly stationary stochastic process $\{X_1,X_2,\ldots\}$. Let $k$ and $l$ be natural numbers and define $y_i = 0$ or 1 according as $x_1 + \cdots + x_{i+k-1}$ is even or odd. Then, for $1 \leq j \leq l$ set $S_j(x_1 \cdots x_n) = \sum_{0\leq i \leq m - 1}y_{j+il}$. We consider all processes that have $(S_1,\ldots,S_l)$ as sufficient statistics. We obtain explicit formulas for the distributions of the processes that are extreme points. We also represent these processes as finitary processes and use this representation to investigate their pairwise independence, ergodicity and mixing properties.