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In probability theory, the law of large numbers (LLN) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. [1]
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]
Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. [1]
LNH was Dirac's personal response to a set of large number "coincidences" that had intrigued other theorists of his time. The "coincidences" began with Hermann Weyl (1919), [2] [3] who speculated that the observed radius of the universe, R U, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:
It is the notion of convergence used in the strong law of large numbers. The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to ...
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]
The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. n → ∞. Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the weak law of large numbers.
By 1689, he had published important work on infinite series and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704.