On stochastic comparison of random vectors

1987 ◽  
Vol 24 (1) ◽  
pp. 123-136 ◽  
Author(s):  
J. George Shanthikumar
Keyword(s):  
Partial Order ◽  
Order Statistics ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Comparison ◽  
Stochastic Monotonicity ◽  
Random Vectors ◽  
Standard Construction ◽  
Frequency Functions ◽  
Independent And Identically Distributed

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.

On stochastic comparison of random vectors

1987 ◽  
Vol 24 (01) ◽  
pp. 123-136 ◽  
Author(s):  
J. George Shanthikumar
Keyword(s):  
Partial Order ◽  
Order Statistics ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Comparison ◽  
Stochastic Monotonicity ◽  
Random Vectors ◽  
Standard Construction ◽  
Frequency Functions ◽  
Independent And Identically Distributed

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.

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

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
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.

scholarly journals Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Computer Program ◽  
Order Statistics ◽  
Stock Exchange ◽  
Random Variables ◽  
Integral Representations ◽  
Symmetric Distribution ◽  
Random Vectors ◽  
Dependent Random Variables ◽  
Exact Distributions

AbstractIntegral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.

SOLVABLE THREE-DIMENSIONAL RATIONAL CHAOTIC MAP DEFINED BY JACOBIAN ELLIPTIC FUNCTIONS

2007 ◽  
Vol 17 (10) ◽  
pp. 3645-3650 ◽  
Author(s):  
ASUKA ONO ◽  
TOHRU KOHDA
Keyword(s):  
Communication Systems ◽  
Chaotic Map ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Space Curve ◽  
Random Vectors ◽  
Rational Map ◽  
Binary Function ◽  
Jacobian Elliptic Functions

Cryptanalysis needs a lot of pseudo-random numbers. In particular, a sequence of independent and identically distributed (i.i.d.) binary random variables plays an important role in modern digital communication systems. Sufficient conditions have been recently provided for a class of ergodic maps of an interval onto itself: R1 → R1 and its associated binary function to generate a sequence of i.i.d. random variables. In order to get more i.i.d. binary random vectors, Jacobian elliptic Chebyshev rational map, its derivative and second derivative which define a Jacobian elliptic space curve have been introduced. Using duplication formula gives three-dimensional real-valued sequences on the space curve onto itself: R3 → R3. This also defines three projective onto mappings, represented in the form of rational functions of xn, yn, zn. These maps generate a three-dimensional sequence of i.i.d. random vectors.

scholarly journals On perpetuities with gamma-like tails

2018 ◽  
Vol 55 (2) ◽  
pp. 368-389 ◽  
Author(s):  
Dariusz Buraczewski ◽  
Piotr Dyszewski ◽  
Alexander Iksanov ◽  
Alexander Marynych
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Exponential Moments ◽  
The One ◽  
The Right ◽  
Independent And Identically Distributed

Abstract An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We provide three disjoint groups of sufficient conditions which ensure that the right tail of a perpetuity ℙ{X > x} is asymptotic to axce-bx as x → ∞ for some a, b > 0, and c ∈ ℝ. Our results complement those of Denisov and Zwart (2007). As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. We give several examples in which the distributions of perpetuities are explicitly identified.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (02) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of N n , the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

scholarly journals Exact laws for sums of ratios of order statistics from the Pareto distribution

2006 ◽  
Vol 4 (1) ◽  
pp. 1-4 ◽  
Author(s):  
André Adler
Keyword(s):  
Order Statistics ◽  
Pareto Distribution ◽  
Random Variables ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Independent And Identically Distributed

AbstractConsider independent and identically distributed random variables {X nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X n(i) ≤ X n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R ij = X n(j)/X n(i).

scholarly journals Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors

2021 ◽  
Vol 9 (1) ◽  
pp. 385-393
Author(s):  
Mhamed Mesfioui ◽  
Julien Trufin
Keyword(s):  
Order Statistics ◽  
Sufficient Conditions ◽  
Random Vectors ◽  
Dispersive Order ◽  
Extreme Order Statistics ◽  
Preservation Property

Abstract In this paper, we investigate sufficient conditions for preservation property of the dispersive order for the smallest and largest order statistics of homogeneous dependent random vectors. Moreover, we establish sufficient conditions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of different sizes.

Limit laws for kth order statistics from conditionally mixing arrays of random variables

1986 ◽  
Vol 23 (03) ◽  
pp. 679-687
Author(s):  
W. Dziubdziela
Keyword(s):  
Weak Convergence ◽  
Order Statistics ◽  
Poisson Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Compound Poisson Distribution ◽  
Limit Laws ◽  
Compound Poisson ◽  
Mixed Compound

We present sufficient conditions for the weak convergence of the distributions of thekth order statistics from a conditionally mixing array of random variables to limit laws which are represented in terms of a mixed compound Poisson distribution.

scholarly journals Recurrence properties of Processes with stationary independent increments

1964 ◽  
Vol 4 (2) ◽  
pp. 223-228 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Increments ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

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