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...</math> almost everywhere, then <math>\mu_n \to \mu</math> in the sense of probability distributions.<ref name ="ito" /> ...ito">{{cite book |last=Itô |first=Kiyosi |year=1984 |title=Introduction to Probability Theory |publisher=Cambridge University Press |page=87 |isbn=0 521 26960 1}} ...

...s boundary that contains all ''N'' points. Wendel's theorem says that the probability is<ref>{{citation |last = Wendel|first = James G.|title = A Problem in Geometric Probability|journal = Math. Scand.|volume = 11|year = 1962|pages = 109–111|url = http:/ ...

...y theory]], the '''Chung–Erdős inequality''' provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The lower bound is ...he form given above by Petrov (in,<ref>{{Cite book|title=Limit theorems of probability theory : sequences of independent random variables|last=Petrov|first=Valent ...

In [[probability theory]], the '''dimension doubling theorems''' are two results about the [[Hausdorff dimension]] of an [[image (mathema == Dimension doubling theorems == ...

{{About|the theorem in probability|the theorem in electricity|Bartlett's bisection theorem}} ...te ''E'' for some particular part of the system and ''p''(''s'',''t'') the probability that a customer who arrives at time ''s'' is in ''E'' at time ''t''. Then t ...

In the mathematical theory of probability, the '''Heyde theorem'''&nbsp;is the [[characterization (mathematics)|chara ...<math>\xi_j, j = 1, 2, \ldots, n, n \ge 2</math>&nbsp; be [[independence (probability theory)|independent]] random variables. Let <math>\alpha_j, \beta_j</math>& ...

...lity measure]]s. The theorem makes a statement about when one can extend a probability measure to a larger [[σ-algebra]]. It is of particular interest for infinit ...st1=Ascherl |first2=Jürgen |last2=Lehn |title=Two principles for extending probability measures |journal=Manuscripta Math. |number=21 |pages=43–50 |date=1977|volu ...

...nach space. Let ''&mu;'' be a centred Gaussian measure on ''X'', i.e. a [[probability measure]] defined on the [[Borel set]]s of ''X'' such that, for every [[bou ...'', ''&mu;'' (equivalently, any ''X''-valued random variable ''G'' whose [[probability distribution|law]] is ''&mu;'') has [[moment (mathematics)|moments]] of all ...

{{Short description|Statement in probability theory}} In the [[mathematics|mathematical]] [[probability theory|theory of probability]], the '''Hsu–Robbins–Erdős theorem''' states that if <math>X_1, \ldots ,X_ ...

{{Short description|Theory in probability theory}} In [[probability theory]], the '''Feldman–Hájek theorem''' or '''Feldman–Hájek dichotomy''' ...

...[probability distribution]] ''P'' admits (in the convolution semi-group of probability distributions) a factorization where ''P''₁ is a probability distribution without any [[indecomposable distribution|indecomposable]] fac ...

...of a probability distribution''' accordingly states that it is the only [[probability distribution]] that satisfies specified conditions. More precisely, the mod ...in measurable metric space <math>(V,d_{v})</math>. By characterizations of probability distributions we understand general problems of description of some set <ma ...

{{Short description|Theorem in probability theory}} ...f>{{Cite journal|last=Rukhin A. L.|date=1970|title=Certain statistical and probability problems on groups|journal=Trudy Mat. Inst. Steklov|volume=111|pages=52–109 ...

where ''P''(A) is the probability that a linear order extending the partial order <math>\prec</math> has the ...he condition that <math>x\prec y</math>. In the language of [[conditional probability]], ...

...right)\sim D_p\left(a_1,\ldots,a_r;a_{r+1}\right)</math>, if their joint [[probability density function]] is == Theorems == ...

In [[probability theory]], '''Kolmogorov's two-series theorem''' is a result about the conve ...>, and we will see that <math>\limsup_N S_N - \liminf_NS_N = 0</math> with probability 1. ...

{{Short description|Formula in probability theory}} ...tes that if ''X'', ''Y'', and ''Z'' are [[random variable]]s on the same [[probability space]], and the [[covariance]] of ''X'' and ''Y'' is finite, then ...

'''Komlós' theorem''' is a theorem from [[probability theory]] and [[mathematical analysis]] about the [[Cesàro summation|Cesàro Let <math>(\Omega,\mathcal{F},P)</math> be a [[probability space]] and <math>\xi_1,\xi_2,\dots</math> be a sequence of real-valued ran ...

...t(a,b\right)</math> (sometimes <math>B_p^I\left(a,b\right)</math>). The [[probability density function]] for <math>U</math> is: {{Probability distribution| ...

{{ about|the theorem in Markov probability theory|the theorem in electrical engineering|Foster's reactance theorem}} In [[probability theory]], '''Foster's theorem''', named after [[Gordon Foster]],<ref>{{Cite ...