On the a.s. Cesàro-α convergence for stationary or orthogonal random variables

We give a new direct proof of the a.s. convergence of the Cesàro- means of a stationary process (X _n) when 0<<1 andE(X _n ^p )<+ with p>1 and we show that this result does not hold in general for p=1. We also consider similar questions for orthogonal random variables. Finally, we study the a.s. convergence of Riesz harmonic means.     

Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes

Let (X_i,Y _i) be a set of observations form a stationary -mixing process and (x) be the conditional -th quantile of Y given X = x. Several authors considered nonparametric estimation of (x) in the i.i.d. setting. Assuming the smoothness of FF(x), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.     

Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed – monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed – semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1–6.     

Identical mixing rates

Summary For strictly stationary sequences, -mixing at a given mixing rate satisfying a log-convexity condition, does not imply -mixing at any essentially faster rate.This work was partially supported by NSF grant MCS 81-01583     

Bounds for optimalα-β binary trees

In this paper we consider a special kind of binary trees where each right edge is associated with a positive number and each left edge with a positive number( ). Given, and the number of nodesn, an optimal tree is one which minimizes the total weighted path length. An algorithm for constructing an optimal tree for given, , n is presented, based on which bounds for balances and total weighted path lengths of optimal trees are derived.     

On the smoothness of conjugacy for circle covering maps

A conjugacy betweenC ¹⁺ -hyperbolic circle covering maps isC ¹⁺ if it has a positive derivative on some point in the -set.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

On some classes of modules

The aim of this paper is to investigate quasi-corational, comonoform, copolyform and -(co)atomic modules. It is proved that for an ordinal a right R-module M is -atomic if and only if it is -coatomic. And it is also shown that an -atomic module M is quasi-projective if and only if M is quasi-corationally complete. Some other results are developed.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

Cesàro α-Integrability and Laws of Large Numbers I

For a sequence of integrable random variables, we introduce a new set of conditions called Cesàro -Integrability and Strong Cesàro -Integrability and show that, for <1/2, these conditions that are strictly weaker than Cesàro Uniform Integrability and Strong Cesàro Uniform Integrability respectively, are sufficient for WLLN and SLLN to hold for a sequence of pairwise independent random variables. For some special kinds of dependent sequences of random variables also, Cesàro -integrability for appropriate is shown to be sufficient for WLLN to hold.