scholarly journals Multivariate Distributions with Fixed Marginals and Correlations

2015 ◽  
Vol 52 (02) ◽  
pp. 602-608
Author(s):  
Mark Huber ◽  
Nevena Marić
Keyword(s):  
Upper Bound ◽  
Convex Combination ◽  
Random Variables ◽  
Multivariate Distributions ◽  
Marginal Distributions ◽  
Four Dimensions ◽  
Bernoulli Random Variables ◽  
Maximal Correlation ◽  
Hoeffding Bounds ◽  
Convexity Parameter

Consider the problem of drawing random variates (X 1, …, X n ) from a distribution where the marginal of each X i is specified, as well as the correlation between every pair X i and X j . For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between X i and X j . Any achievable correlation between X i and X j is a convex combination of these bounds. We call the value λ(X i , X j ) ∈ [0, 1] of this convex combination the convexity parameter of (X i , X j ) with λ(X i , X j ) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F 1, …, F n of (X 1, …, X n ), we show that λ(X i , X j ) = λ ij if and only if there exist symmetric Bernoulli random variables (B 1, …, B n ) (that is {0, 1} random variables with mean ½) such that λ(B i , B j ) = λ ij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

scholarly journals Multivariate Distributions with Fixed Marginals and Correlations

2015 ◽  
Vol 52 (2) ◽  
pp. 602-608 ◽  
Author(s):  
Mark Huber ◽  
Nevena Marić
Keyword(s):  
Upper Bound ◽  
Convex Combination ◽  
Random Variables ◽  
Multivariate Distributions ◽  
Marginal Distributions ◽  
Four Dimensions ◽  
Bernoulli Random Variables ◽  
Maximal Correlation ◽  
Hoeffding Bounds ◽  
Convexity Parameter

Consider the problem of drawing random variates (X1, …, Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj. Any achievable correlation between Xi and Xj is a convex combination of these bounds. We call the value λ(Xi, Xj) ∈ [0, 1] of this convex combination the convexity parameter of (Xi, Xj) with λ(Xi, Xj) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F1, …, Fn of (X1, …, Xn), we show that λ(Xi, Xj) = λij if and only if there exist symmetric Bernoulli random variables (B1, …, Bn) (that is {0, 1} random variables with mean ½) such that λ(Bi, Bj) = λij. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

scholarly journals SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS

2019 ◽  
Vol 7 ◽  
Author(s):  
ASAF FERBER ◽  
VISHESH JAIN
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Upper Bound ◽  
Matrix Theory ◽  
Random Variables ◽  
The Other ◽  
Symmetric Matrices ◽  
Combinatorial Approach ◽  
Bernoulli Random Variables ◽  
Small Constant

Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$ . On the other hand, the best known upper bound on the singularity probability of $M_{n}$ , due to Vershynin (2011), is $2^{-n^{c}}$ , for some unspecified small constant $c>0$ . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_{n}$ is at most $2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large $n$ . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.

scholarly journals An Upper Bound on the Size of Avoidance Couplings

2018 ◽  
Vol 28 (3) ◽  
pp. 325-334
Author(s):  
ERIK BATES ◽  
LISA SAUERMANN
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Random Variables ◽  
Bernoulli Random Variables ◽  
Random Walkers

We show that a coupling of non-colliding simple random walkers on the complete graph on n vertices can include at most n - log n walkers. This improves the only previously known upper bound of n - 2 due to Angel, Holroyd, Martin, Wilson and Winkler (Electron. Commun. Probab.18 (2013)). The proof considers couplings of i.i.d. sequences of Bernoulli random variables satisfying a similar avoidance property, for which there is separate interest.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

scholarly journals Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

2011 ◽  
Vol 02 (11) ◽  
pp. 1382-1386 ◽  
Author(s):  
Deepesh Bhati ◽  
Phazamile Kgosi ◽  
Ranganath Narayanacharya Rattihalli
Keyword(s):  
Random Variables ◽  
Weighted Sum ◽  
Bernoulli Random Variables

Dependent Random Variables with Independent Subsets - II

1990 ◽  
Vol 33 (1) ◽  
pp. 24-28 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Marginal Distributions ◽  
Dependent Random Variables ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

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