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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 law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function √ n log log n, intermediate in size between n of the law of large numbers and √ n of the central limit theorem, provides a non-trivial limiting behavior.
The law of truly large numbers (a statistical adage), attributed to Persi Diaconis and Frederick Mosteller, states that with a large enough number of independent samples, any highly implausible (i.e. unlikely in any single sample, but with constant probability strictly greater than 0 in any sample) result is likely to be observed. [1]
This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers, as it is less rigorous and general than the modern version. [27] After these four primary expository sections, almost as an afterthought, Bernoulli appended to Ars Conjectandi a tract on calculus, which concerned infinite series. [16]
Law of total probability; Law of large numbers; Bayes' theorem; ... An extension of the addition law to any number of sets is the inclusion–exclusion principle.
The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the German mathematician Leopold Kronecker .
The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem.There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S n, scaled by n −1, converge to zero, respectively in probability and almost surely:
For every (fixed) x, F n (x) is a sequence of random variables which converge to F(x) almost surely by the strong law of large numbers. That is, F n converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of F n to F by the Glivenko–Cantelli theorem. [2]