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Book review

Author(s) Galambos, Janos (ed.)
Kátai, Imre (ed.)
Title Probability Theory and Applications
Publisher Kluwer Academic Publishers
Year of publication 1992
   
Reviewed by P.P. Zabrejko

This book is a collections of 22 original articles written by leading and we]]-known scientists in probability theory and its applications. The spectrum of topics presented in this book is extremely broad and one can find here numerous new ideas and results in theory of inequalities, limit theorems, renewal theory and reliability theory, characterizations of distributions, infinite divisibility of polynomials of normal variables, limiting distributions for order statistics, stochastic processes, functional equations in engineering model building, probabilistic number theory and many others. The book is dedicated to the memory of Professor József Mogyoródi (1933-1990) of L. Eötvös University, Budapest, Hungary, the leading and well-known scientist in probability theory and adjoining fields and remarkable educator in mathematics, who unexpectedly died during a scientific visit to Holland; the themes in the main part of the articles in the book covers most fields investigated by Mogyoródi.

The article by L. Takács, Random walk processes and their various applications, deals with Bernoulli excursion and tied-down random walk which are considered as functions of a random variable defined as the area of a random set bounded by the real line and the sample function; as applications of obtained results the distribution and the asymptotic distribution of the total height of a random tree are presented as well as the asymptotic behaviour of the moments and the distribution of a statistic which measures the deviation between two empirical distribution functions. The article by N.L. Bassily, Duality of the Birkholder-Davis-Gundy inequality and the generalized Feffermann-Garsia inequality II, presents a theorem on duality of the generalized Fefferman-Garsia inequality and the Birholder-Davis-Gundy inequality each to other in Orlicz spaces.

The article by F. Weisz, Martingale Hardy spaces with continuous time, describes some new results about atomic decompositions of martingale Hardy spaces, and the martingale inequalities and corresponding duality theorems between Hardy and BMO spaces.

The article by Gy. Michaletzky, Construction of optimal Hankel approximations in the guise of stochastic processes, presents a new approach to investigate the classical problem about approximation of rational function <formula> with stable matrix A with an analogous function <formula> with stable F but having fixed number of poles inside the unit circle. The article by M. Arató, Lévy's random domains on the plain, proposes analytical proof of Lévy's heuristic statement on the distribution of the directed random domain for complex AR processes. The article by V.K. Rohatgi, G.J. Székely, On the Infinite divisibility of polynomials in infinitely divisible random variables, presents the proof of infinitely divisibility of polynomials of the second degree, a counterexample for polynomials of the third degree and theorem on the linearity of polynomials of random variable with support {0, 1, 2, ...}.

The article by J. Galambos, Random sample sizes: limit theorems and characterizations, presents some curious results on statistics with random samples sizes that are based on consideration instead of an original probabilistic model its extension in which random sample size is introduced as variable. The article by A. Kováts, T. Móri, Aging solutions of certain renewal type equations, is devoted to analysis of the equation S = pW + qS*V with unknown distribution S and given non-negative distributions W and V, p + q = 1; as applications some probabilistic modes are considered. The articles by J. Galambos, Lee Min-Young, Extensions of some Bonferroni-type Inequalities to multivariable setting, and by J. Galambos, Xu Yuan, Univariate and multivariate Bonferroni-type Inequalities: methods for proof and questions of optimality, some aspects of Bonferoni-type inequalities are discussed; in particular, a row of the generalizations of Bonferoni-type inequality for the joint distribution of the number of occurrences in several sequence of events under natural suppositions. The article by Masaaki S., Sharp Bonferroni-type inequalities in explicit forms, presents sharp Bonferroni type estimates of the probabilities that exact or at least m out of n events occur in terms (Sk, ..., Sr) where Sj is the sum of probabilities of simultaneous realization j out of n events. The article by L. Szeidl, V.M. Zolotarev, Analytical representation of limit distributions for a class of random symmetric polynomials, deals with an analytical representation of the density of the limit distribution function as <formula> for the sequence of homogeneous symmetric polynomials of n random variables with a common distribution function of some special type. The article by H. Dress, R.-D. Reiss, Tail behavior in Wicksell's corpuscle problem, is devoted to the shapes of upper tails of distributions of 'sphere radii' and 'circle radii' that are connected by the Wicksell integral transformation.

The article by F. Schipp, Universal contractive projections and a.e. convergence, presents a generalization of the Dor-Odell-Burkholder inequality for a nondecreasing sequence of contractive projections in Lebesgue space <formula>, on to 'sequences' of such projections that are indexed by elements of a tree. The article by P. Deheuvels, Pointwise Bahadur-Kiefer-type theorem I, is devoted to approximation problem of the Bahadur-Kiefer process by some analytical expressions from independent Wiener processes on (-, ). The article by M. Falk, F. Marohn, Laws of small numbers: some applications to conditional curve estimation, is devoted to probabilistic analysis of rare events (to laws of small numbers) that is based on the investigation of first and second steps Poisson process approximation and to applications to regression analysis for nonparametric and semiparametric cases. The article by E. Castillo, A. Fernandez-Cantelli, R. Ruiz Cobo, Design of statistical lifetime models by functional equations, is devoted to discussion of three statistical models of the fatigue life of longitudinal elements. The article by L. Lakatos, V. Ceric, Another approach to the ergodic distribution in the M/G/1 system, presents some new method to find the ergodic distribution in the M/G/1 queueing system.

The article by K.-H. Indekofer, A new method in probabilistic number theory, presents a new probabilistic approach to problems of number theory that is based on the Stone-Cech compactification N* of N and the corresponding integration theory for almost summable functions on N*; the new theory allows to consider classical arithmetic functions (including the Möbius function mu) and new types of functions that are important in number theory (for example, almost multiplicative functions). The article by I. Kátai, Distribution of Q-additive functions, deals with functions <formula> for which the decomposition <formula> implies the equality <formula>.

The article by K.-H. Indekofer, I. Kátai, P. Recskó, Number systems and fractal geometry, continues analysis some probabilistic properties of number systems and in particular deals with complete additive functions with respect to number systems. The article by Z. Daróczy, A. Járai, T. Szabó, On sequences of solid type, presents the existence and uniqueness theorem in l1 for sequences satisfying the recursion relation <formula>.

One can see that this book is more closely to a monograph than a collection of articles. Moreover this book can serve as the basis for a row of special mathematical courses to graduate and post-graduate students. Undoubtedly this book is useful for specialist in probability theory and mathematical statistics but not only - this book is interesting for specialists in analysis and in many mathematical fields where analytical methods play essential part.