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).     

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

Abstract

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||V _n || ≤ b _n } (where V _n is a random element of a separable Banach space) is played by events of the form {X _n  ? B _n } (where X _n is a random closed bounded set).     

General Weak Laws of Large Numbers for Bootstrap Sample Means

Abstract

For bootstrap sample means resulting from a sequence {X _n , n ≥ 1} of random variables, very general weak laws of large numbers are established. The random variables {X _n , n ≥ 1} do not need to be independent or identically distributed or be of any particular dependence structure. In general, no moment conditions are imposed on the {X _n , n ≥ 1}. Examples are provided that illustrate the sharpness of the main results.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

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)     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Prophet Regions for Discounted,Uniformly Bounded Random Variables

Abstract

Let X ₁, X ₂,… be any sequence of [0,1]-valued random variables. A complete comparison is made between the expected maximum E(max _j≤n Y _j ) and the stop rule supremum sup _t E Y _t for two types of discounted sequences: (i) Y _j  = b _j X _j , where {b _j } is a nonincreasing sequence of positive numbers with b ₁ = 1; and (ii) Y _j  = B ₁… B _j?1 X _j , where B ₁, B ₂,… are independent [0,1]-valued random variables that are independent of the X _j , having a common mean β. For instance, it is shown that the set of points {(x, y): x = sup _t E Y _{{(x, y): x=sup t E Y} and y = E(max _j≤n Y _j ), for some sequence X ₁,…,X _n and Y _j  = b _j X _j }, is precisely the convex closure of the union of the sets {(b _j x, b _j y): (x, y) ∈ C _j }, j = 1,…,n, where C _j  = {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ x[1 + (j ? 1)(1 ? x ^1/(j?1))]} is the prophet region for undiscounted random variables given by Hill and Kertz [8 Hill , T.P. , and R.P. Kertz . 1983 . Stop rule inequalities for uniformly bounded sequences of random variables . Trans. Amer. Math. Soc. 278 : 197 – 207 . [CSA]  [Google Scholar]]. As a special case, it is shown that the maximum possible difference E(max _j≤n β ^j?1 X _j ) ? sup _t E(β ^t?1 X _t ) is attained by independent random variables when β ≤ 27/32, but by a martingale-like sequence when β > 27/32. Prophet regions for infinite sequences are given also.     

Minimal graded free resolutions for monomial curves in 𝔸4 defined by almost arithmetic sequences

Let m = (m₀, m₁, m₂, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m₀, m₁, m₂, n) = 1, such that m₀ < m₁ < m₂ form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m₀? +m₁? +m₂? +n?. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X₀ = t^m₀, X₁ = t^m₁, X₂ = t^m₂, Y = tⁿ is a graded R-module, where R is the polynomial ring k[X₀, X₁, X₂, Y] with the grading degX_i: = m_i, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

Let {X _n ;n≥1} be a sequence of i.i.d. random variables and let X ^(r) _n = X _j if |X _j | is the r-th maximum of |X ₁|, ..., |X _n |. Let S _n = X ₁+⋯+X _n and ^(r) S _n = S _n −(X ⁽¹⁾ _n +⋯+X ^(r) _n ). Sufficient and necessary conditions for ^(r) S _n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for ^(r) S _n are studied. Received March 22, 1999, Revised November 6, 2000, Accepted March 16, 2001     

Generalized One-Sided Laws for the Larger Observations of a Triangular Array

Consider independent and identically distributed random variables {X,X _nj , 1jn,n1} with density f(x)=px ^–p–1 I(x1), where p>0. We show that there exist unusual generalized Laws of the Iterated Logarithm involving the larger order statistics from our array.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

On B(4, 13) 2-Groups

A group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j  | 1 ≤ i, j ≤ n}| ≤k. In this article, we give a complete characterization of B(4, 13) 2-groups, and then obtain a complete characterization of B(4, 13) groups.     

Limsup results and LIL for partial sum processes of a Gaussian random field

Let {ξ _j ; j ∈ ℤ₊ ^d be a centered stationary Gaussian random field, where ℤ₊ ^d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ^d, having nonnegative integer coordinates. For each j = (j ₁ , ..., jd) in ℤ₊ ^d , we denote |j| = j ₁ ... j _d and for m, n ∈ ℤ₊ ^d , define S(m, n] = Σ _m<j≤n ζ _j , σ²(|n−m|) = ES ² (m, n], S _n = S(0, n] and S ₀ = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa     

Chains of Rings with Local Formal Fibers

Let (T, M) be a complete local domain containing the integers. Let p ₁ ? p ₂ ? ··· ? p _n be a chain of nonmaximal prime ideals of T such that T _{p n} is a regular local ring. We construct a chain of excellent local domains A _n  ? A _n?1 ? ··· ? A ₁ such that for each 1 ≤ i ≤ n, the completion of A _i is T, the generic formal fiber of A _i is local with maximal ideal p _i , and if I is a nonzero ideal of A _i then A _i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht _T Q, then there is a chain of excellent local domains B ₀ ? B ₁ ? ··· ? B _h  ? T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B _i ) = i, the completion of B _i is isomorphic to T[[X ₁, X ₂,…, X _i ]] where the X _j 's are indeterminants, and the formal fiber of Q ∩ B _i is local.     

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ₁, δ₁] ··· [Θ _n , δ _n ] an iterated differential operator k-algebra over R such that δ _j (Θ _i ) ∈ R[Θ₁, δ₁] ··· [Θ _i?1, δ _i?1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A _P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ _i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.     

A Remark on Self-Normalization for Dependent Random Variables

Let (X _t , t ∈ Z) be a stationary process, and let S _n = ∑_{1⩽ i⩽n} X _i . In this paper, we consider the central limit theorem for the self-normalized sequence S _n /U _n , where U _n ² = ∑_1⩽j⩽N Y _j ² , Y _j = ∑_{(j−1)m<i⩽jm} X _i , n = mN. We show how such a self-normalization works for AR(1) and MA(q) processes.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 173–183, April–June, 2005.