Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product
نویسندگان
چکیده
منابع مشابه
On the Borel-Cantelli Lemma
In the present note, we propose a new form of the Borel-Cantelli lemma.
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The notation and terminology used here have been introduced in the following papers: [17], [3], [4], [8], [13], [1], [2], [5], [15], [14], [21], [9], [12], [11], [16], [6], [20], [19], and [18]. For simplicity, we adopt the following rules: O1 is a non empty set, S1 is a σ-field of subsets of O1, P1 is a probability on S1, A is a sequence of subsets of S1, and n is an element of N. Let D be a s...
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The Borel-Cantelli Lemma is very important in the probability theory. In this paper, we first describe the general case of the Borel-Cantelli Lemma. The first part of this lemma, assuming convergence and the second part includes divergence and independence assumptions. In the following, we have brought generalizations of the first and second part of this lemma. In most generalizat...
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The celebrated Borel-Cantelli lemma asserts that (A) If ~P(E,,) < 00, then P (lim sup Ek) =O; (B) If the events Es are independent and if xP(Ek) = m, then P(lim sup Eh) = 1. In intuitive language P(lim. sup &) is the probability that the events Eh occur “infinitely often” and will be denoted by P(Ek i.0.). This lemma is the basis of all theorems of the strong type in probability theory. Its app...
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1973
ISSN: 0091-1798
DOI: 10.1214/aop/1176996800