theory of probability, a Borel right process, named after Émile Borel, is a particular kind of continuous-time random process. Let E {\displaystyle E}...

theorem Borel right process Borel set Borel summation Borel distribution Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture...

The Borel distribution is a discrete probability distribution, arising in contexts including branching processes and queueing theory. It is named after...

theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco...

\left(S,\Sigma _{S}\right)} is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse...

{\displaystyle i\in I} . Consider a stochastic process X : [0, T] × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets...

} Itô processes were first named due to their role in this theorem, though Itô had previously studied them. Markov process Borel right process Gilbert...

\mathbb {R} ^{d})} , with its Borel σ {\displaystyle \sigma } -algebra, such that: 1. (Initial Condition) The process starts at ξ {\displaystyle \xi...

Stationary Stochastic Processes for Scientists and Engineers. CRC Press. p. 11. ISBN 978-1-4665-8618-5. Aumann, Robert (December 1961). "Borel structures for...

] ) {\displaystyle \mathrm {Borel} ([0,t])} be the Borel sigma algebra on [ 0 , t ] {\displaystyle [0,t]} . The process X {\displaystyle X} is said to...

closed (or more precisely, Borel measurable) region B {\textstyle B} of the plane. The number of points of a point process N {\displaystyle \textstyle...

algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra b {\displaystyle {\mathfrak {b}}}...

topology. The set of all such strings forms a sigma algebra, specifically, a Borel algebra. This algebra is then commonly written as ( Ω , B ) {\displaystyle...

Sg=\{s\cdot g\,:\,s\in S\}.} Left and right translates map Borel sets onto Borel sets. A measure μ {\displaystyle \mu } on the Borel subsets of G {\displaystyle...

also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. Markov showed that the law can apply...

In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect...

measure Eξ (also known as mean measure) of a point process ξ is a measure on S that assigns to every Borel subset B of S the expected number of points of...

To denote the number of points of N {\displaystyle {N}} located in some Borel set B {\displaystyle B} , it is sometimes written Φ ( B ) = # ( B ∩ N )...

-\delta _{0}))\,} where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by exp ⁡ ( ν ) = ∑ n = 0 ∞ ν ∗ n n ...

x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\right)^{-{\frac {1}{k}}}=\infty } for every bounded Borel A ⊆ Λ. The eigenvalues of a random m × m Hermitian...

{\displaystyle \varepsilon _{t}} appear on the right side of the equation. The autocorrelation function of an AR(p) process can be expressed as [citation needed]...

on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular...

{\displaystyle f:\left[a,b\right]\rightarrow \mathbb {R} }   is Borel-measurable and bounded and   g : [ a , b ] → R {\displaystyle g:\left[a,b\right]\rightarrow \mathbb...

the use of the "monkey metaphor" is that of French mathematician Émile Borel in 1913, but the first instance may have been even earlier. Jorge Luis Borges...

from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M....

\pi ^{-1}(x)\right)=0,} and so μ x ( E ) = μ x ( E ∩ π − 1 ( x ) ) {\displaystyle \mu _{x}(E)=\mu _{x}(E\cap \pi ^{-1}(x))} ; for every Borel-measurable...

measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections...

point process being interpreted as a random set. Alternatively, the number of points of N {\displaystyle \textstyle {N}} located in some Borel set B {\displaystyle...

doi:10.1016/j.neuroscience.2012.05.032. PMID 22626643. S2CID 23629664. Borel, L.; Lopez, C.; Péruch, P.; Lacour, M. (Dec 2008). "Vestibular syndrome:...

some Borel set B is often written as: N ( B ) , {\displaystyle \textstyle {N}(B),} which reflects a random measure interpretation for point processes. These...