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It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century. [1] [2] A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have ...
This is a direct implication from the Borel–Cantelli lemma. If S n is a sum of n real independent random variables: = + + then S n converges almost surely if and only if S n converges in probability. The proof can be found in Page 126 (Theorem 5.3.4) of the book by Kai Lai Chung. [13]
Proof: We will prove this statement using the portmanteau lemma, part A. First we want to show that (X n, c) converges in distribution to (X, c). By the portmanteau lemma this will be true if we can show that E[f(X n, c)] → E[f(X, c)] for any bounded continuous function f(x, y). So let f be such arbitrary bounded continuous function.
Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the ...
The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X. Using the discrete metric. The Borel–Cantelli lemma is an example application of these constructs. Using either the discrete metric or the Euclidean metric
Borel–Cantelli lemma; Doob–Dynkin lemma; Itô's lemma (stochastic calculus) ... An example of a covering described by the Knaster–Kuratowski–Mazurkiewicz lemma.
The concept of a normal number was introduced by Émile Borel . Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers. Wacław Sierpiński showed that it is possible to specify a particular such number.
Borel–Cantelli lemma; C. Covering lemma; ... Vitali covering lemma; W. Whitney covering lemma This page was last edited on 1 January 2018, at 13:47 (UTC) ...