Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.
Stochastic is the property of being well-described by a random probability distribution. Stochasticity and randomness are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a stochastic process is also referred to as a random process.
In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochastic processes that by definition possess independent increments are the Wiener process, all Lévy processes, all additive process and the Poisson point process.
Albert Nikolayevich Shiryaev is a Soviet and Russian mathematician. He is known for his work in probability theory, statistics and financial mathematics.
Anatoliy Volodymyrovych Skorokhod was a Soviet and Ukrainian mathematician.
In statistics, an exchangeable sequence of random variables is a sequence X1, X2, X3, ... whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.
Bálint Tόth is a Hungarian mathematician whose work concerns probability theory, stochastic process and probabilistic aspects of mathematical physics. He obtained PhD in 1988 from the Hungarian Academy of Sciences, worked as senior researcher at the Institute of Mathematics of the HAS and as professor of mathematics at TU Budapest. He holds the Chair of Probability at the University of Bristol and is a research professor at the Alfréd Rényi Institute of Mathematics, Budapest.
Harry Kesten was a Jewish American mathematician best known for his work in probability, most notably on random walks on groups and graphs, random matrices, branching processes, and percolation theory.
Jan Hendrik van Schuppen is a Dutch mathematician and Professor at the Department of Mathematics of the Vrije Universiteit, known for his contributions in the field of systems theory, particularly on control theory and system identification, on probability, and on a number of related practical applications.
Geoffrey Richard Grimmett is an English mathematician known for his work on the mathematics of random systems arising in probability theory and statistical mechanics, especially percolation theory and the contact process. He is the Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge, and was the Master of Downing College, Cambridge, from 2013 to 2018.
Rollo Davidson was a probabilist, alpinist, and Fellow-elect of Churchill College, Cambridge, who died aged 25 on Piz Bernina. He is known for his work on semigroups, stochastic geometry, and stochastic analysis, and for the Rollo Davidson Prize, given in his name to early-career probabilists.
Carl-Gustav Esseen was a Swedish mathematician. His work was in the theory of probability. The Berry–Esseen theorem is named after him.
Carl Svante Janson is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987.
Probability Theory and Related Fields is a peer-reviewed mathematics journal published by Springer. Established in 1962, it was originally named Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, with the English replacing the German starting from volume 71 (1986). The journal publishes articles on probability. The journal is indexed by Mathematical Reviews and Zentralblatt MATH. Its 2019 MCQ was 2.29, and its 2019 impact factor was 2.125.
Bruce Edward Hajek is a Professor in the Coordinated Science Laboratory, the head of the Department of Electrical and Computer Engineering, and the Leonard C. and Mary Lou Hoeft Chair in Engineering at the University of Illinois Urbana–Champaign. He does research in communication networking, auction theory, stochastic analysis, combinatorial optimization, machine learning, information theory, and bioinformatics.
Peter Jagers is a Professor Emeritus of Mathematical Statistics at University of Gothenburg and Chalmers University of Technology who made lasting contributions in probability and general branching processes. Jagers was first vice president (2007–2010) of the Royal Swedish Academy of Sciences and Chair of the Royal Society of Arts and Sciences in Gothenburg (2012). He in an elected member of the International Statistical Institute, a Fellow of the Institute of Mathematical Statistics and past President of the Bernoulli Society (2005–2007). He also served as a member of the Scientific Advisory Board of Statistics Sweden.
In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.
