Areas Of Probability Theory Research Articles

Inequalities for order statistics of samples from distributions with monotone hazard rate

A number of inequalities relating the mathematical expectations of order statistics or functions of order statistics in the case where the sample is taken from a distribution with monotone hazard rate function are presented. These inequalities can be applied in reliability theory and other areas of probability theory and mathematical statistics. In particular, in the paper, they are used to obtain new characterizations of the exponential distribution.

B.V. Gnedenko: Classic of Limit Theorems in the Theory of Probability

B.V. Gnedenko is an outstanding scientist-mathematician, who worked in the area of probability theory and its applications. B.V. Gnedenko deserved world-wide popularity by his investigations of limit distributions for sums of independent random variables (Gnedenko and Kolmogorov 1954) that completed a long period of probability theory development up to the middle of XX-th century. The main idea of “accompanied infinitely divisible distributions” developed by B.V. Gnedenko, became a guidance in the limit theory of semimartingales as it is presented by Jacod and Shiryaev (1987) and others (Cinlar et al. 1980). The triplet of predictable characteristics for semimartingale is the main idea in investigation the limit behavior for the random evolutions in the scheme of Poisson approximation (Koroliuk and Limnios, Theory Probab Appl 49(4):629–644, 2005b).

Retail Real Estate Cycles as Markov Chains

Executive Summary. Commercial real estate markets across cities evolve with considerable risk with respect to variables such as rent levels and growth, occupancy levels and trends, capital market developments, and construction conditions. The complex patterns are more understandable with the help of the real estate cycle models reviewed here. Describing the future with a set of alternative cycle points and their anticipated probabilities is an application of a well-developed area of probability theory, Markov chain analysis. Numerical examples show how descriptions of cycle conditions in an initial period can be used to generate risk formatted descriptions of cycle conditions. The results are plausible descriptions of future risks. Standard Markov chain calculations allow the real estate cycle analyst to anticipate the proportion of quarters that a market will reside in each cycle point.

Pólya’s Theorem on Random Walks via Pólya’s Urn

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsDavid A. LevinDAVID A. LEVIN (Ph.D. University of California, Berkeley, 1999) is an assistant professor of mathematics at the University of Oregon. His research interests are in probability theory, and he is a co-author of the bookMarkov Chains and Mixing Times (AMS, 2008).Yuval PeresYUVAL PERES (Ph.D. Hebrew University, Jerusalem, 1990) is a principal researcher and manager of the Theory Group at Microsoft Research in Redmond. He is also an adjunct professor at the University of California, Berkeley and at the University of Washington, Seattle. He is the author of more than 150 research papers and co-author of the bookMarkov Chains and Mixing Times (AMS, 2008). His research interests include most areas of probability theory, as well as parts of ergodic theory, game theory, and information theory. He received the Rollo Davidson and Loeve prizes, and was an invited speaker at the 2002 International Congress of Mathematics (Beijing) and the 2008 European Congress of Mathematics (Amsterdam).

On the 70th Birthday of D. M. Chibisov

December 25, 2006, was the 70th birthday of the outstanding mathematician in the area of probability theory and mathematical statistics, Doctor of Physical and Mathematical Sciences, Professor, member of the Editorial Board of our journal, DMITRII MIKHAILOVICH CHIBISOV. The Editorial Board of Theory of Probability and Its Applications cordially congratulates Dmitrii Mikhailovich on his anniversary and wishes him sound health and further success in his creative work.

The XVIIIth International Seminar on Stability Problems for Stochastic Models

The XVIIIth Seminar on Stability Problems for Stochastic Models was held from January 26 to February 1, 1997, in a small town, Hajduszoboszlo, situated a few kilometers from Debrecen, Hungary. Traditionally, the seminar was held under the aegis of the Mathematical Institute of the RAS. The seminar was supported by the Kossuth University of Debrecen (Hungary), the Moscow State University (Russia), the Hungarian Bernoulli Society, the Hungarian National Foundation for Scientific Research, and the “Catching Up with European Higher Education” Fund. The seminar was attended by more than 100 specialists from Belgium, Bulgaria, Hungary, Germany, Holland, Greece, Israel, India, Lithuania, Poland, Russia, Ukraine, France, Switzerland, Yugoslavia, and Japan. The subjects presented at the seminar included all areas of probability theory and mathematical statistics traditionally discussed at the seminars on stability problems for stochastic models: limit theorems in probability theory and mathematical statistics, characterization of probability distributions and their stability, the theory of probability, metrics, limit theorems, and characterization problems in stochastic analysis, renewal theory, queueing theory, metrics in mathematical statistics, applied statistics. Along with these themes some reports were devoted to problems of actuarial and financial mathematics. The present seminar had one significant peculiarity. It was dedicated to the 60th anniversary of V. M. Zolotarev — founder, organizer, and head of the international seminars on stability problems for stochastic models — and M. Arato — an active participant in all the seminars and the organizer of some recent seminars. In their scientific reports and informal speeches at the anniversary party, the participants noted the outstanding contribution of V. M. Zolotarev and M. Arato to modern probability theory. The seminar was held in a congenial and friendly atmosphere. This was possible due to the efforts of the Hungarian organizers of the seminar, who provided remarkable conditions for fruitful scientific discussions and informal contacts between the participants. The present seminar differs in no way from the XVIth seminar which was also held in Hungary (in Eger) from August 29 to September 3, 1994, with the very active participation of more than 100 mathematicians from 20 countries. We are convinced that Hungary is hospitable at any season of the year. Unfortunately, the proceedings of the XVIth Seminar, as well as the Proceedings of the XVIIth Seminar, held in Kazan’ June 19–26, 1995 (more than 70 participants from 15 countries), have not been published in Russian. They were published in the Journal of Mathematical Sciences (Vol. 76, No. 2, 1995; Vol. 78, No. 1, 1996; Vol. 81, No. 5, 1996; Vol. 83, No. 3, 1997). In this issue of Theory of Probability and its Applications we publish a brief summary of the topics at the XVIIth Seminar on Stability Problems for Stochastic Models which evoked the greatest interest. The proceedings of the seminar will be published in three issues of the Journal of Mathematical Sciences. The next XIXth International Seminar on Stability Problems for Stochastic Models will be held in September 1998 in Vologda (Russia).

Remarks on the life and research of Roland L. Dobrushin

The life and research work of Professor R.L. Dobrushin (1929‐1995) had a profound influence on several areas of probability theory, information theory, and mathematical physics. The paper contains a biographical note, a review of Dobrushin′s results, and a list of his publications.

Brownian excursions on combs

In this paper we use techniques from Ito excursion theory to analyze Brownian motion on generalized combs. Ito excursion theory is a little-known area of probability theory and we therefore present a brief introduction for the uninitiated. A general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined. We then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs.