Displaying similar documents to “Gaussian random band matrices and operator-valued free probability theory”

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

Hiroyuki Okazaki, Yasunari Shidama (2010)

Formalized Mathematics

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In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.

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.

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...

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...