On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

On a Lower Bound of L p Norms in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive lower bounds of the L _p norms _np for all p, 1p, in the central limit theorem for -mixing random variables with finite sixth-order moments in a strictly stationary case and finite eighth-order moments in a not necessarily stationary one.     

Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and León (Extremes 3(1):57–86, 2000), Kratz and León (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and León in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincaré characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.     

Entropy Theorem for Random Fields on Trees

We study the entropy theorem or the asymptotic equipartition property (AEP)for random fields on bomogeneous trees.A tree is a graph which is connected and contains no circuits. We discuss mainlya homogeneous tree T on which each vertex has N neighboring vertices. T is bipartitegraph because we can partition its vertex set into two equivalence classes, where α～β     

Convergence Properties for Arrays of Rowwise φ-Mixing Random Variables

Convergence properties for arrays of rowwise(φ-mixing random variables are studied.As an application,the Chung-type strong law of large numbers for arrays of rowwiseφ-mixing random variables is obtained.Our results extend the corresponding ones for independent random variables to the case of φ-mixing random variables.     

Strong Laws and Summability for φ-Mixing Sequences of Random Variables

In this note the almost sure convergence of stationary, -mixing sequences of random variables according to summability methods is linked to the fulfillment of a certain integrability condition generalizing and extending the results for i.i.d. sequences. Furthermore we give via Baum-Katz type results an estimate for the rate of convergence in these laws.     

On the Strong Law of Large Numbers for φ-Sub-Gaussian Random Variables

Ukrainian Mathematical Journal - For p ≥ 1, let φp(x) = x2/2 if |x| ≤ 1 and φp(x) = 1/p|x|p ? 1/p + 1/2 if |x| > 1. For a random variable ξ, let $$...     

Local Central Limit Theorem for Multi-group Curie–Weiss Models

Journal of Theoretical Probability - We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this...     

Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

Let (X _i ) _i1 be an i.i.d. sequence of random elements in the Banach space B, S _n X ₁++X _n and _n be the random polygonal line with vertices (k/n,S _k ), k=0,1,...,n. Put (h)=h L(1/h), 0h1 with 0<1/2 and L slowly varying at infinity. Let H ⁰ (B) be the Hölder space of functions x:[0,1]B, such that x(t+h)–x(t)=o((h)), uniformly in t. We characterize the weak convergence in H ⁰ (B) of n ^–1/2 _n to a Brownian motion. In the special case where B= and (h)=h , our necessary and sufficient conditions for such convergence are E X ₁=0 and P(|X ₁|>t)=o(t ^–p()) where p()=1/(1/2–). This completes Lamperti (1962) invariance principle.     

The Pólya Sum Process: Limit Theorems for Conditioned Random Fields

Zessin (J. Contemp. Math. Anal. 44(1):36–44, 2009) constructed the so-called Pólya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the Pólya sum process as was done for the Poisson process by Nguyen and Zessin (Z. Wahrscheinlichkeitstheor. Verw. Geb. 37(3):191–200, 1976/77).     

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X _i ) be a stationary and ergodic Markov chain with kernel Q and f an L ² function on its state space. If Q is a normal operator and f=(I?Q)^1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L ²), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lif?ic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\), in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\), q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(I?Q)^1/2 L ², or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c _n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\), the CLT need not hold.     

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.     

Cesàro Summation for Random Fields

Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence of the Cesàro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by ℤ₊², the positive two-dimensional integer lattice points.     

Kolmogorov–Chentsov Theorem and Differentiability of Random Fields on Manifolds

A version of the Kolmogorov–Chentsov theorem on sample differentiability and Hölder continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.     

Large Deviations for the Boundary Crossing Probabilities of Some Random Fields

For Y_1,Y_2,…i.i.d.with Y_1～N(μ,1) and S_n=sum from 1 to n(Y_i),the large deviations are obtained for the probabilities that conditionally given (i) S_m=0, and (ii) S_(m_i)=Applied these results to the double change-points model with some nuisance parameters, we developed the large deviation for the significance level of the likelihood ratio test.     

The Law of the Iterated Logarithm for the Sums of φ-Mixing Sequences with Duple Suffixes

Hu Shuhe gets a sufficient condition on the law of the iterated logarithm for the sums of φ-mixing sequences with duple suffixes. This paper greatly improves his condition.     

On the Improved Form of the Bounded Theorem for Pseudo-Differential Operators

Caldern A.P.et al.were studied a class of the bounded pseudo-differential operators of order -M and type ρ,δ_1,δ_2, and showed them to be bounded in L~2 provided that 0≤ρ≤δ_1<1, 0≤ρ≤δ_2<1 and M/n≥1/2(δ_1+δ_2)-ρ.Hrmander et al.have shown that if the aforecited conditions are not valid, then the conclusion that A is bounded in L~2 isn't necessarily right. Caldern et al. settled the borderline case and remove the restriction on the support of α.     

Komlós Theorem for Unbounded Random Sets

We present the Komlós theorem for multivalued functions whose values are closed (possibly unbounded) convex subsets of a separable Banach space. Komlós theorem can be seen as a generalization of the SLLN for it deals with a sequence of integrable multivalued functions that do not have to be identically distributed nor independent. The Artstein–Hart SLLN for random sets with values in Euclidean spaces is derived from the main result. Finally, since the main theorem concerns multifunctions whose values are allowed to be unbounded, we can restate it in terms of normal integrands (random lower semicontinuous functions).     

On the Hochstadt–Lieberman Theorem for Discontinuous Boundary-valued Problems

In this paper, we discuss the half inverse problem for Sturm–Liouville equations with boundary conditions dependent on the spectral parameter and a finite number of discontinuities inside the interval and prove the Hochstadt–Liberman type theorem for the above boundary-valued problem.     

On the Skitovich–Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f _n: XX a homomorphism f _n(x)=nx, and X ⁽ⁿ⁾=Im f _n. Denote by I(X) the set of idempotent distributions on X and by (X) the set of Gaussian distributions on X. Consider linear statistics L ₁= ₁( ₁)+ ₂( ₂) and L ₂= ₁( ₁)+ ₂( ₂), where _j are independent random variables taking on values in X and with distributions _j, and _j, _jAut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L ₁ and L ₂ implies that ₁, ₂I(X) if and only if X possesses the property: for each prime p the factor-group X/X ^(p) is finite. If X is connected, then there exist independent random variables _j taking on values in X and with distributions _j, and _j, _jAut(X) such that L ₁ and L ₂ are independent, whereas ₁, ₂(X) * I(X).