Multivariate Convex Orderings, Dependence, and Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Download Full-text

On the law of the iterated logarithm in the infinite variance case

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021868 ◽

1980 ◽

Vol 30 (1) ◽

pp. 5-14 ◽

Cited By ~ 2

Author(s):

R. A. Maller

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Infinite Variance ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient ◽

Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

Download Full-text

Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems

Mathematics ◽

10.3390/math8050749 ◽

2020 ◽

Vol 8 (5) ◽

pp. 749 ◽

Cited By ~ 1

Author(s):

Yury Khokhlov ◽

Victor Korolev ◽

Alexander Zeifman

Keyword(s):

Limit Theorems ◽

Probability Distributions ◽

Sufficient Conditions ◽

Stable Distributions ◽

Necessary And Sufficient Conditions ◽

Independent Random Variables ◽

Random Vectors ◽

Scale Mixtures ◽

Elliptically Contoured ◽

Necessary And Sufficient

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution.

Download Full-text

Stability of maxima of random variables defined on a Markov chain

Advances in Applied Probability ◽

10.2307/1426000 ◽

1972 ◽

Vol 4 (2) ◽

pp. 285-295 ◽

Cited By ~ 13

Author(s):

Sidney I. Resnick

Keyword(s):

Markov Chain ◽

Regularity Condition ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Finite Markov Chain ◽

Finite Product ◽

Normalizing Constants ◽

Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

Download Full-text

Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails

Science China Mathematics ◽

10.1007/s11425-010-4012-9 ◽

2010 ◽

Vol 53 (6) ◽

pp. 1421-1434 ◽

Cited By ~ 44

Author(s):

Li Liu

Keyword(s):

Heavy Tails ◽

Sufficient Conditions ◽

Random Variables ◽

Moderate Deviations ◽

Necessary And Sufficient Conditions ◽

Dependent Random Variables ◽

Necessary And Sufficient

Download Full-text

The law of the iterated logarithm for exchangeable random variables

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171295000482 ◽

1995 ◽

Vol 18 (2) ◽

pp. 391-396

Author(s):

Hu-Ming Zhang ◽

Robert L. Taylor

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Exchangeable Random Variables ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient

In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.

Download Full-text

Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5049 ◽

2021 ◽

Vol 37 ◽

pp. 359-369

Author(s):

Marko Kostadinov

Keyword(s):

Linear Combination ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Linear Combinations ◽

Special Cases ◽

Sufficient And Necessary Conditions ◽

Alpha Beta ◽

Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

Download Full-text

An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

10.36227/techrxiv.14987115 ◽

2021 ◽

Author(s):

Yann Donnelly ◽

Michael Peter Kennedy

Keyword(s):

Nonlinear Distortion ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Quantization Noise ◽

Linear Combinations ◽

Upper Limit ◽

Comprehensive Theory ◽

Free Behavior ◽

Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

Download Full-text

Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

Advances in Applied Probability ◽

10.1017/s0001867800016189 ◽

1986 ◽

Vol 18 (04) ◽

pp. 865-879 ◽

Cited By ~ 3

Author(s):

Svante Janson

Keyword(s):

Random Walks ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Partial Sums ◽

Exit Times ◽

First Passage ◽

Partial Sum ◽

Positive Expectation ◽

Necessary And Sufficient

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation. We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

Download Full-text

On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

Journal of Applied Probability ◽

10.1239/jap/1032374230 ◽

1999 ◽

Vol 36 (1) ◽

pp. 78-85 ◽

Cited By ~ 1

Author(s):

M. S. Sgibnev

Keyword(s):

Markov Chains ◽

Random Walks ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Stationary Distributions ◽

Necessary And Sufficient ◽

Submultiplicative Function ◽

Independent Identically Distributed

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

Download Full-text