The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

Large deviations viewpoints for a heavy-tailed β-mixing sequence

Let (Xn;n≥1) be a stationary sequence of non-negative random variables with heavy tails. Under mixing conditions, we study logarithmic asymptotics for the distributions of the partial sums sn=X1 X2 … Xn. We obtain the crude estimates P(Sn>nx)≈n-αx 1 for appropriate values of x, where a is a specific parameter. The related conjecture proposed by Gantert is investigated. As a by-product, the so-called supremum large deviations principle is also studied.     

Large deviations viewpoints for a heavy-tailed β-mixing sequence

Let {Xn; n ≥ 1} be a stationary sequenceof non-negative random variables with heavy tails. Under mixing conditions, we study logarithmic asymptotics for the disp(Sn ＞ nx) ≈ n-αx+1 for appropriate values of x, where α is a specific parameter. The related conjecture proposed by Gantert is investigated. As a by-product, the so-called supremum large deviations principle is also studied.     

The vanishing rate of Weil–Petersson sectional curvatures

Geometriae Dedicata - A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new...     

The convergence of moments in the central limit theorem for stationary ϕ-mixing processes

Пусть {X_j} - строго стац ионарная последоват ельностьс ?перемешиванием, EXj-Q,E¦-X _j¦^r<∞ для некоторогоr>2. Положим \(S_n = \mathop \sum \limits_{j = 1}^n X_j \) . Ибрагимов (1962) доказал, что если приn →∞, то 1 $$\mathop {\lim }\limits_{n \to \infty } P\{ S_n /\sigma _n< x\} = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^x e^{{{ - u^2 } \mathord{\left/ {\vphantom {{ - u^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} du.$$ В работе установлено, что при указанных выш е условиях в этой центральной пр едельной теореме имеет место т акже и сходимостьr-ых абсолютных моментов, т.е. если σ _n ² →∞ приn→ ∞, то $$\mathop {\lim }\limits_{n \to \infty } E|S_n /\sigma _n |^r = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^{ + \infty } |u|^r e^{ - u^2 /2} du.$$ Этот результат обобщ ает один более ранний результат автора (1980 г.).     

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 the strong law of large numbers for ��-mixing and ��-mixing random variables

We consider the classical Kolmogorov condition for strong law of large numbers for sequences of dependent random variables; the so-called ??-mixing and Rademacher?CMenchoff condition for ??-mixing sequences.     

Geometric and combinatorial structure of a class of spherical folding tessellations — I

A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed.     

The quality of bank forecasts: The dollar-pound exchange rate, 1990–1993

A general framework for examining the quality of currency forecasts is described. This framework incorporates existing accuracy measures and modifies them to give components of accuracy that are appropriate for statistically derived and judgementally based forecasts. The framework is described and applied to one week ahead US$/UK£ forecasts from three major banks over a three year period between 1990 and 1993. The results suggest that, while overall forecast performance was poor, some aspects of the predictions could still be useful in a practical setting.     

Oracle Inequality for Sparse Trace Regression Models with Exponential β-mixing Errors

In applications involving,e.g.,panel data,images,genomics microarrays,etc.,trace regression models are useful tools.To address the high-dimensional issue of these applications,it is common to assume some sparsity property.For the case of the parameter matrix being simultaneously low rank and elements-wise sparse,we estimate the parameter matrix through the least-squares approach with the composite penalty combining the nuclear norm and the l₁ norm.We extend the existing analysis of th...     

The asymptotic variance rate of the output process of finite capacity birth-death queues

We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ ^*+∑v _i where λ ^* is the rate of outputs and v _i are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of v _i are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity. In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal, is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced. Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.     

Strong Laws of Large Numbers for Weighted Sums of Ч-mixing Sequence

In this paper, strong laws of large numbers for weighted sums of Ч-mixing sequence are investigated. Our results extend the corresponding results for negatively associated sequence to the case of Ч-mixing sequence.     

Complete Convergence for Arrays of Rowwise Ч-mixing Random Variables

In the paper, the complete convergence for arrays of rowwise Q-mixing random variables is studied. Some sufficient conditions for complete convergence for an array of row wise Q-mixing random variables without assumptions of identical distribution and stochastic domination are presented.     

The Laplacian on β-sets via the method of averages

In this paper, we show how the symmetric Laplacian on β-sets with $0<\beta <\frac{1}{2}$, together with its associated Dirichlet form and harmonic functions, can be defined entirely in terms of average values of a function over basic sets. We have generalized Strichartz’s and our previous results by using the method of averages.     

Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables, and the Baum-Katz-type result for arrays of rowwise φ-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of φ-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).     

Limiting behavior for arrays of rowwise ρ *-mixing random variables

We study the limiting behavior of maximal partial sums for arrays of rowwise ?? ^*-mixing random variables and obtain some new results that improve the corresponding theorem of Zhu [M.H. Zhu, Strong laws of large numbers for arrays of rowwise ?? ^*-mixing random variables, Discrete Dyn. Nat. Soc., 2007, Article ID 74296, 6 pp., 2007].