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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 ...
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
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]
Download QR code; Print/export Download as PDF; Printable version; In other projects Wikidata item; Appearance. ... Borel–Cantelli lemma; Brunn–Minkowski theorem; C.
Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken. Note that it suffices to prove the result for a small interval I = (− ε , ε ), since if ψ ( t ) is a smooth bump function with compact support in (− ε , ε ) equal ...
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.
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
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 ( 1917 ) showed that it is possible to specify a particular such number.