Maximal inequalities for demimartingales and a strong law of large numbers

Chow's maximal inequality for (sub)martingales is extended to the case of demi(sub)martingales introduced by Newman and Wright (Z. Wahrsch. Verw. Geb. 59 (1982) 361–371). This result serves as a “source” inequality for other inequalities such as the Hajek–Renyi inequality and Doob's maximal inequality and leads to a strong law of large numbers. The partial sum of mean zero associated random variables is a demimartingale. Therefore, maximal inequalities and a strong law of large numbers are obtained for associated random variables as special cases.     

Maximal inequalities and their applications to orthogonal and Hadamard matrices

Periodica Mathematica Hungarica - Maximal inequalities for the signed vector summands are established. Probabilistic estimations for the sets of appropriate signs are given. By use of the...     

Maximal inequalities for not necessarily orthogonal random variables and some applications

Статья посвящена обо бщению неравенства Меньшова-Радемахера. Условие ортогональн ости заменяется усло вием (4), т.е. предполагается, что с уществует функционалh, зависящ ий от совместного рас пределения случайных величинX _m+1 , Х _m+2 ,..., Х _m+n , который субаддити вен в смысле условия (5) и удовлетворяет услов ию (4). Из этих допущений выводится неравенство вида (6). В ч астном случае Ф(х)=¦х¦^p(р≧1) в нер авенстве (6) мы имеемB _Φ (y)=y ^p . Во второй части стать и из указанных нераве нств выводятся некоторые усиленные законы больших чисел. Типичн ым результатом являе тся следующий. Пусть р>1 и x ₁, Х₂,... —после довательность случа йных величин. Если для всех m≧0, n≧1 выполнены неравенс тва $$E\left( {\left| {\mathop \Sigma \limits_{i = m + 1}^{m + n} X_i } \right|^p } \right) \leqq h(F_{m,n} )$$ где функционал h(F _m,n )удо влетворяет условию (5) u $$h(F_{0,n} ) = O\left( {\frac{{n^p }}{{(\log n)^{p + 1} (\log \log n)^2 }}} \right)(n \to \infty )$$ , mo $$P(S_n /n \to 0) = 1$$ .     

Correlation inequalities and their applications

Correlation inequalities and their applications to the question of the presence or absence of phase transitions is classical lattice systems are considered. A major part of the known inequalities is obtained as a corollary of a single general theorem.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 16, pp. 5–37, 1979.     

Some convolution inequalities and their applications

We introduce a class of convolution inequalities and study the implications of these inequalities for certain problems in harmonic analysis.

     

Maximal inequalities,convexity inequality and their duality. I

Для положительных су бмартингалов А. М. Гарс иа [2] и автор [3] установили максимал ьное неравенство в случае общей функции Юнга в предположении, что до полнительная по Юнгу функция удовлет воряет так называемо му условию роста, т. е. что степень дополнительной функции конечна. Этот результат обобщает классическое нераве нство Дуба. В данной работе изуча ются подобные максим альные неравенства, но в случ ае, когда уже сама функция Юнга имеет конечную степе нь. Дается характеристика функ ций Юнга конечной степени, для которых может быть ус тановлено максимальное нераве нство.     

Functional inequalities for transient birth-death processes and their applications

We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth-death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth-death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.     

Some generalized integral inequalities and their applications

In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations.     

On retarded integral inequalities and their applications

The main results of [J. Math. Anal. Appl. 285 (2003) 436-443] are here generalized to the following retarded integral inequalities:

     

Systems of convex inequalities and their applications

In this paper, we shall prove some results related to doubly-stochastic operators and invariant measures which may be obtained from a general condition by Fan (Math. Z. 68 (1957), 205–217) for the existence of solutions of general systems of convex inequalities in topological vector spaces.     

Maximal inequalities for the Ornstein-Uhlenbeck process

Let be the Ornstein-Uhlenbeck velocity process solving

with , where 0$"> and is a standard Brownian motion. Then there exist universal constants 0$">and 0$"> such that

for all stopping times of . In particular, this yields the existence of universal constants 0$"> and 0$"> such that

for all stopping times of . This inequality may be viewed as a stopped law of iterated logarithm. The method of proof relies upon a variant of Lenglart's domination principle and makes use of Itô calculus.

     

Maximal inequalities,convexity inequality,and their duality. II

, , . . . [1], , . , , ., , L logL. , , . . . . [5]. , .     

Maximal inequalities for a continuous semimartingale

Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 .     

Some new delay integral inequalities and their applications

The aim of the present paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations.     

Chebyshev type inequalities involving permanents and their applications

We extend the well-known Chebyshev’s inequality to some new cases involving permanents under the proper hypotheses. Our main results are

     

Some new nonlinear difference inequalities and their applications

In this paper, we establish some new nonlinear difference inequalities in two independent variables, which can be used as handy tools in the study of qualitative properties of solutions of certain classes of difference equations.     

Polynomial inequalities on measurable sets and their applications

We study Pólya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.