Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

Hiroyuki Okazaki; Yasunari Shidama

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Probability on Finite Set and Real-Valued Random Variables

Hiroyuki Okazaki, Yasunari Shidama (2009)

Formalized Mathematics

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In the various branches of science, probability and randomness provide us with useful theoretical frameworks. The Formalized Mathematics has already published some articles concerning the probability: [23], [24], [25], and [30]. In order to apply those articles, we shall give some theorems concerning the probability and the real-valued random variables to prepare for further studies.

Random Variables and Product of Probability Spaces

Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

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We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of...

Monotone increasing multi-valued condensing random operators and random differential inclusions.

Dhage, B.C., Ntouyas, S.K., Palimkar, D.S. (2006)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

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Random walks with $k$ -wise independent increments.

Benjamini, Itai, Kozma, Gady, Romik, Dan (2006)

Electronic Communications in Probability [electronic only]

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Convergence results and sharp estimates for the voter model interfaces.

Belhaouari, S., Mountford, T., Sun, Rongfeng, Valle, G. (2006)

Electronic Journal of Probability [electronic only]

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The 123 theorem of Probability Theory and Copositive Matrices

Alexander Kovačec, Miguel M. R. Moreira, David P. Martins (2014)

Special Matrices

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Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral...

Elementary Introduction to Stochastic Finance in Discrete Time

Peter Jaeger (2012)

Formalized Mathematics

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This article gives an elementary introduction to stochastic finance (in discrete time). A formalization of random variables is given and some elements of Borel sets are considered. Furthermore, special functions (for buying a present portfolio and the value of a portfolio in the future) and some statements about the relation between these functions are introduced. For details see: [8] (p. 185), [7] (pp. 12, 20), [6] (pp. 3-6).