Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n ^−1/2 β _n log ^3/2 n in which β _n can be particularly taken as (log log n)^1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n ^−1/2(log log n)^1/2log ² n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n ^−1/2 β _n log ^1/2 n for the above β _n under the given covariance function, which improves the relevant one n ^−1/2(log log n)^1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

On Estimates of Factorial Quotients

We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n ⁻¹(2n)!!²/(2n − 1)!!² → π. We improve the Wallis inequalities (n + 1/2)⁻¹(2n)!!²/(2n − 1)!!² < π < n ⁻¹(2n)!!²/(2n − 1)!!² for π and obtain new estimates of factorial quotients with error order not worse than 1/n ². __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005.     

Topology of Random 2-Complexes

We study the Linial–Meshulam model of random two-dimensional simplicial complexes. One of our main results states that for p≪n ⁻¹ a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π ₁(Y) is free and H ₂(Y)=0, asymptotically almost surely. Our other main result gives a precise threshold for collapsibility of a random 2-complex to a graph in a prescribed number of steps. We also prove that, if the probability parameter p satisfies p≫n ^−1/2+ϵ , where ϵ>0, then an arbitrary finite two-dimensional simplicial complex admits a topological embedding into a random 2-complex, with probability tending to one as n→∞. We also establish several related results; for example, we show that for p<c/n with c<3 the fundamental group of a random 2-complex contains a non-abelian free subgroup. Our method is based on exploiting explicit thresholds (established in the paper) for the existence of simplicial embeddings and immersions of 2-complexes into a random 2-complex.     

On cardinalities of row spaces of Boolean matrices

We prove that there are exactlyn numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2 ⁿ⁻¹+1,2 ⁿ⁻¹+2,2 ⁿ⁻¹+4, ..., 2 ⁿ⁻¹+2 ⁿ⁻², 2 ⁿ . Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2 ⁿ⁻¹+n−1. The second is that the numbers listed above are precisely the numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of topologies on a finite setX withn elements.     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

Growth and recurrence of stationary random walks

 Let (X _n ,n≥1) be a real-valued ergodic stationary stochastic process, and let (Y _n =X ₁ +…+X _n ,n≥1) be the associated random walk. We prove the following: if the sequence of distributions of the random variables Y _n /n,n≥1, is uniformly tight (or, more generally, does not have the zero measure as a vague limit point), then there exists a real number c such that the random walk (Y _n −nc,n≥1) is recurrent. If this sequence of distributions converges to a probability measure ρ on ℝ (or, more generally, has a nonzero limit ρ in the vague topology), then (Y _n −nc,n≥1) is recurrent for ρ−a.e.cℝ. Received: 24 September 2001 / Revised version: 1 August 2002 / Published online: 24 October 2002 The first author was partially supported by the FWF research project P14379-MAT. Mathematics Subject Classification (2000): 37A20, 37A50, 60G10, 60G50 Key words or phrases: Recurrent stationary random walks – Recurrent cocycles     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Testing Epidemic Changes of Infinite Dimensional Parameters

To detect epidemic change in the mean of a sample of size n of random elements in a Banach space, we introduce new test statistics DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length log^γ n, γ > 1. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrices of random vectors. Final version 27 October 2004     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.     

On The Correlation Of Binary M-sequences

We obtain the upper bound O(2^14n/15 n^−1/5) on the number of distinct values of all possible correlation functions between M-sequences of order n .     

Efficient algorithms for heavy-tail analysis under interval uncertainty

Most applications of statistics to science and engineering are based on the assumption that the corresponding random variables are normally distributed, i.e., distributed according to Gaussian law in which the probability density function ρ(x) exponentially decreases with x: ρ(x)∼exp (−k⋅x ²). Normal distributions indeed frequently occur in practice. However, there are also many practical situations, including situations from mathematical finance, in which we encounter heavy-tailed distributions, i.e., distributions in which ρ(x) decreases as ρ(x)∼x ^−α . To properly take this uncertainty into account when making decisions, it is necessary to estimate the parameters of such distributions based on the sample data x ₁,…,x _n —and thus, to predict the size and the probabilities of large deviations. The most well-known statistical estimates for such distributions are the Hill estimator H for α and the Weismann estimator W for the corresponding quantiles.     

A note on the Bahadur representation of sample quantiles for α-mixing random variables

By some moment inequalities for α-mixing random variables, we prove the Bahadur representation of sample quantiles under very weak α-mixing coefficients. As application, the uniformly asymptotic normality is derived, the rate of which is near to n ^−1/6 under the given conditions.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}ⁿ, where nearest-neighbor edges are occupied with probability p. Let p_c(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2^n/3, where λ is a small positive constant. Let ε=n(p−p_c(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2^−n/3) is of size Θ(2^2n/3), below this scaling window it is at most 2(log 2)nε⁻², and above this scaling window it is at most O(ε2ⁿ). In this paper, we prove that for the size of the largest component is at least Θ(ε2ⁿ), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

On large deviations and density functions

Suppose thatX ₁,X ₂, ... is a sequence of absolutely continuous or integer valued random variables with corresponding probability density functionsf _n (x). Let {φ _n } _n=1 ^∞ be a sequence of real numbers, then necessary and sufficient conditions are given forn ⁻¹ logf _n (φ _n )-n ⁻¹ log P (X _n >φ _n )=0(1) asn→∞.     

A characterization of limiting distributions of estimators in an autoregressive process

Summary Let the random variablesX ₁,X ₂, ...,X _n be generated by the first-order autoregressive modelX _i =θX _i−1 +e _i wheree _i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ², and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.