Max-infinite divisibility and multivariate total positivity

1994 ◽  
Vol 31 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Abdulhamid A. Alzaid ◽  
Frank Proschan
Keyword(s):  
Distribution Function ◽  
Total Positivity ◽  
Positive Function ◽  
Infinite Divisibility ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Positive Dependence ◽  
Totally Positive

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

Max-infinite divisibility and multivariate total positivity

1994 ◽  
Vol 31 (03) ◽  
pp. 721-730 ◽  
Author(s):  
Abdulhamid A. Alzaid ◽  
Frank Proschan
Keyword(s):  
Distribution Function ◽  
Total Positivity ◽  
Positive Function ◽  
Infinite Divisibility ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Positive Dependence ◽  
Totally Positive

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

scholarly journals SECOND ORDER SUBEXPONENTIALITY AND INFINITE DIVISIBILITY

2020 ◽  
pp. 1-24 ◽  
Author(s):  
TOSHIRO WATANABE
Keyword(s):  
Asymptotic Behaviour ◽  
Real Line ◽  
Infinite Divisibility ◽  
Second Order ◽  
Divisible Distribution ◽  
Infinitely Divisible Distribution ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Exponential Moment ◽  
The Real

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$ th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

scholarly journals Covariance Matrix Estimation under Total Positivity for Portfolio Selection*

2020 ◽  
Author(s):  
Raj Agrawal ◽  
Uma Roy ◽  
Caroline Uhler
Keyword(s):  
Covariance Matrix ◽  
Total Positivity ◽  
General Purpose ◽  
Shrinkage Estimators ◽  
Positive Dependence ◽  
Matrix Estimation ◽  
Sample Covariance ◽  
Totally Positive ◽  
Markowitz Portfolio ◽  
Previous State

Abstract Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

Max-infinite divisibility

1977 ◽  
Vol 14 (02) ◽  
pp. 309-319 ◽  
Author(s):  
A. A. Balkema ◽  
S. I. Resnick
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Extremal Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

scholarly journals Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

2016 ◽  
Vol 27 (04) ◽  
pp. 1650037 ◽  
Author(s):  
Mingchu Gao
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Triangular Array ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Space Operator ◽  
Infinitely Divisible ◽  
Operator Model

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

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