The mixing advantage for bounded random variables

Corresponding to n independent non-negative random variables X₁,…,X_n concentrated on a bounded interval set are values M₁,…,M_n, where each M_i is the expected value of the maximum of n independent copies of X_i. We obtain a sharp upper bound for the expected value of the maximum of X₁,…,X_n in terms of M₁,…,M_n. This inequality is sharp. A similar result is demonstrated for minima.     

Inequalities for sums of independent geometrical random variables

Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the functionF(p ₁,...,p_n;t)=P(X₁+...+X_n≤t) is Schur-concave with respect to (p ₁,...,p_n) for every realt, whereX _i are independent geometric random variables with parametersp _i. A generalization to negative binomial random variables is also presented.     

The expected length of the longest probe sequence for bucket searching when the distribution is not uniform

We study the expected value of the maximum number of accesses needed to locate an element in a hashing file constructed by using an order-preserving hashing function and with collision resolution by the method of separate chaining. It is assumed that X₁, …, X_n are independent [0,1]-valued random variables with common density f, and that X_i is hashed to the $\text{nX}{}_{\mathrm{i}}\text{+ 1}\text{st}$ bucket (chain). For all densities that are bounded, the expected value of the maximum number of accesses is shown to be asymptotic to $\text{log}\text{n}\text{log log}\text{n}$, and the dependency of this expected value on f is made explicit by exhibiting the first few terms in the asymptotic expansion. For unbounded f, a tight upper bound is given for the expected value.     

Three characterizations of non-binary correlation-immune and resilient functions

A functionf(X ₁,X ₂, ...,X _n ) is said to betth-order correlation-immune if the random variableZ=f(X ₁,X ₂,...,X _n ) is independent of every set oft random variables chosen from the independent equiprobable random variablesX ₁,X ₂,...,X _n . Additionally, if all possible outputs are equally likely, thenf is called at-resilient function. In this paper, we provide three different characterizations oft th-order correlation immune functions and resilient functions where the random variable is overGF (q). The first is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays.     

Confidence Bands for Isotonic Median Curves Using Sign Tests

Suppose that one observes independent random variables (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n) in R² with unknown distributions, except that Median(Y_i | X_i = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n²). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.     

On samples of independent random vectors in spaces of infinitely increasing dimension

We study the asymptotic behavior of a set of random vectors ξ_i, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets M _m⁽ⁿ⁾ = ξ₁,..., ξ_m and X _n^λ = x ∈ X _n: ρ(x) ≤ λn, and the mutual location of pairs of vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998.     

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ₃-metric measuring the difference between expectations of sufficiently smooth functions, like |·|³, of a sum of independent random variables X ₁,..., X _n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X _i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X ₁,..., X _n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

Asymptotics of Maxima of Discrete Random Variables

If X₁, X₂,..., X_n are independent and identically distributed discrete random variables and M_n=max (X₁,..., X_n) we examine the limiting behavior of (M_n–b(n))/a(n) as n . It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n , it is possible to obtain non-degenerate limits for (M_n–b(n))/a(n). We consider four families of discrete distributions and show how this can be done.     

Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

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and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.     

Bounds for the Concentration of Asymptotically Normal Statistics

Let X₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X ₁,...,X _n ) for which ET² < . Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.     

Ratio comparisons of supremum and stop rule expectations

Summary Suppose X ₁,X ₂,...,X_n are independent non-negative random variables with finite positive expectations. Let T _n denote the stop rules for X ₁,...,X _n. The main result of this paper is that E(max{X ₁,...,X _n }) <2 sup{EX _t :tT _n }. The proof given is constructive, and sharpens the corresponding weak inequalities of Krengel and Sucheston and of Garling.Partially supported by AFOSR Grant F49620-79-C-0123     

p-Norm bounds on the expectation of the maximum of a possibly dependent sample

Let X₁, X₂,…, X_n be identically distributed possibly dependent random variables with finite pth absolute moment assumed without loss of generality to be equal to 1. Denote the order statistics by X_1:n, X_2:n,…, X_n:n. Bounds are derived for E(X_n:n) when it is assumed that the X_i's are (i) arbitrarily dependent and (ii) independent. The effect of assuming a symmetric common distribution for the X_i's is discussed. Analogous bounds are described for the expected range of the sample. Bounds on expectations of general linear combinations of order statistics are described in the independent case.     

Storage capacity of a dam with gamma type inputs

Summary Consider mutually independent inputsX ₁,...,X _n onn different occasions into a dam or storage facility. The total input isY=X ₁+...+X _n. This sum is a basic quantity in many types of stochastic process problems. The distribution ofY and other aspects connected withY are studied by different authors when the inputs are independently and identically distributed exponential or gamma random variables. In this article explicit exact expressions for the density ofY are given whenX ₁,...,X _n are independent gamma distributed variables with different parameters. The exact density is written as a finite sum, in terms of zonal polynomials and in terms of confluent hypergeometric functions. Approximations whenn is large and asymptotic results are also given.     

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

Consider 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).     

MAXIMUM TREES OF FINITE SEQUENCES

ＭＡＸＩＭＵＭＴＲＥＥＳＯＦＦＩＮＩＴＥＳＥＱＵＥＮＣＥＳ￥ＷＵＳＨＩＱＵＡＮＡｂｓｔｒａｃｔ：Ｌｅｔｎ，ｓ１，ｓ２，．．．，ｓｎｂｅｎｏｎ－ｎｅｇａｔｉｖｅｉｎｔｅｇｅｒｓａｎｄＭ（ｓ１，ｓ２，．．．，ｓｎ）＝｛（ａ１，ａ２，．．．，ａ．）｜ａｉｉ...     

Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables

Let {X ₁,...,X _N} be a set of N independent random variables, and let S _n be a sum of n random variables chosen without replacement from the set {X ₁,...,X _N} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S _n under mild conditions.     

Functions of elementary random variables and their application to system reliability

Summary LetX ₁,...,X _n be elementary random variables, i.e. random variables taking only finitely many values in . Then for an arbitray functionf(X ₁,...,X _n ) inX ₁,...,X _n a unique polynomial representation with the aid of Lagrange polynomials is given. This easily yields the moments as well as the distribution off(X ₁,...,X _n ) by terms of finitely many moments of the variablesX ₁,...,X _n . For n=1 a necessary and sufficient condition results thatm numbers are the firstm moments of a random variable takingm+1 different values. As an application of random functionsf(X ₁,...,X _n ) the reliability of technical systems with three states is treated.

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Zusammenfassung X ₁, ...,X _n seien elementare Zufallsvariable, d. h., Zufallsvariable, die nur endlich viele reelle Werte annehmen. Mit Hilfe von Lagrangepolynomen wird für eine beliebige Funktionf(X₁,...,X _n ) eine eindeutige polynomiale Darstellung angegeben. Daraus ergeben sich leicht die Momente wie auch die Verteilung von f(X₁,...,X _n ), ausgedrückt durch die Momente der VariablenX ₁,...,X _n . Fürn=1 erhält man eine notwendige und hinreichende Bedingung, daßm Zahlen die erstenm Momente einer Zufallsvariablen sind, diem+1 verschiedene Werte annimmt. Als Anwendung wird die Zuverlässigkeit eines technischen Systems mit drei Zuständen behandelt.

     

Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Abstract

Consider independent and identically distributed random variables {X _nk , 1 ≤ k ≤ m, n ≥ 1} from the Pareto distribution. We randomly select a pair of order statistics from each row, X _n(i) and X _n(j), where 1 ≤ i < j ≤ m. Then we test to see whether or not Strong and Weak Laws of Large Numbers with nonzero limits for weighted sums of the random variables X _n(j)/X _n(i) exist where we place a prior distribution on the selection of each of these possible pairs of order statistics.     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)