On numbers with given digit distributions

Archiv der Mathematik -     

Construction of multivariate distributions with given marginals

Summary We make some remarks on the problem how to construct probability measures with given marginals. Questions of this kind arise if one wants to build a stochastic model in a situation where one has some idea of the kind of dependence and knows exactly certain marginal distributions.     

Certain bivariate distributions and random processes connected with maxima and minima

The minimum and the maximum of t independent, identically distributed random variables have \(\bar F^{t}\) and F ^t for their survival (minimum) and the distribution (maximum) functions, where \(\bar F = 1-F\) and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t >?0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by F ^t . However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.     

Limit distributions for the bivariate geometric maxima

Limit laws are established for the behavior of (max X _i , max Y _i ) when (X _i , Y _i ) are independent and distributed according to a bivariate geometric distribution.     

Uncorrelatedness sets for random variables with given distributions

Let and be random variables having finite moments of all orders. The set

is said to be an uncorrelatedness set of and It is known that in general, an uncorrelatedness set can be arbitrary. Simple examples show that this is not true for random variables with given distributions. In this paper we present a wide class of probability distributions such that there exist random variables with given distributions from the class having a prescribed uncorrelatedness set. Besides, we discuss the sharpness of the obtained result.

     

Limiting distributions of maxima under triangular schemes

A well-known result in extreme value theory indicates that componentwise taken sample maxima of random vectors are asymptotically independent under weak conditions. However, in important cases this independence is attained at a very slow rate so that the residual dependence structure plays a significant role.In the present article, we deduce limiting distributions of maxima under triangular schemes of random vectors. The residual dependence is expressed by a technical condition imposed on the spectral expansion of the underlying distribution.     

Selfdecomposable distributions for maxima of independent random vectors

Summary In the present paper the limit laws for conveniently normalized multivariate sample extremes are characterized by means of the decomposability of probability distributions. Continuous automorphisms ofR ^d =[–,]^d with respect to the operation v defined by x y=(max(x _i, y_i),i=1... d) are treated as norming mappings. An integral representation of the limit distributions is found using their log-concavity and a decomposition ofR ^d in orbits of the norming family. Finally an example is given as an illustration.Research supported in part by the Committee of Science, Bulgarian Concil of Ministers, under contract no. 60/1987     

Martingales with sufficient statistics

The property of countable convexity of the set which contains all dominated martingale laws, for a family of measurable functions with values in a separable Borel space, is proved to be equivalent to the existence of a sufficient statistic. The result is then used to derive a representation of such laws in terms of their extreme points, which are laws of uniformly integrable martingales. Finally, we show that martingales with sufficient statistics converge with probability 1.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Expansions and penultimate distributions of maxima of bivariate normal random vectors

It is well known that the marginal maxima of n$n$ standard normal random vectors with correlation coefficient ρ<1$\rho <1$ are asymptotically independent. In this article, the residual dependence will be captured by asymptotic expansions and certain penultimate distributions including the case where ρ(n)↑1$\rho \left(n\right)\uparrow 1$ at a certain rate.     

Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

Martingales and character ratios

Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.

     

Characterizations of distributions by moment properties of sums of records and successive maxima

We consider the estimation problem for expectations of random variables that are sums of records and successive maxima. The corresponding upper bounds and characterizations of source distributions are obtained in the case where these bounds are attained, Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov; POMI, Vol. 361, 2008, pp. 138–144.     

Limit distributions for the maxima of discrete random variables under monotone normalization

Lithuanian Mathematical Journal - Let X1,X2, . . .,Xn be independent and identical discrete random variables, and let Mn = max(X1, . . .,Xn) denote their maximum. There has been considerable work...     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Reverse Martingales and Approximation Operators

Let {ξ_n, _n, n ≥ m ≥ 1} be a reverse martingale such that the distribution of ξ_n depends on x I R =(− ∞, ∞)x. for each n ≥ m, and ξ_n[formula] For a continuous bounded function f on R let L_n(f, x) = Ef(ξ_n) be the associated positive linear operator. The properties of ξ_n are used to obtain the convergence properties of L_n(f, x), and some more details are given when ξ_n is a reverse martingale sequence of -statistics. Lipschitz properties for a subclass of these operators resulting from an exponential Family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.