Gnedenko-Kolmogorov Central Limit Theorem Revisited
Let be a probability density of the random variable, i.e.
, and .
Suppose that all variables are independent.
The mean and the variance of a given random variable are, respectively
.
The mean and variance of are therefore and .
The density of the random variable corresponding to the sum is given by the
Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov) .
Let be distributions such that .
Let, and .
Without loss of generality assume that and .
The random variable holds, as,
where and
Read more about this topic: Beta Wavelet
Famous quotes containing the words central, limit, theorem and/or revisited:
“When life has been well spent, age is a loss of what it can well spare,—muscular strength, organic instincts, gross bulk, and works that belong to these. But the central wisdom, which was old in infancy, is young in fourscore years, and dropping off obstructions, leaves in happy subjects the mind purified and wise.”
—Ralph Waldo Emerson (1803–1882)
“Today, the notion of progress in a single line without goal or limit seems perhaps the most parochial notion of a very parochial century.”
—Lewis Mumford (1895–1990)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“And yet we constantly reclaim some part of that primal spontaneity through the youngest among us, not only through their sorrow and anger but simply through everyday discoveries, life unwrapped. To see a child touch the piano keys for the first time, to watch a small body slice through the surface of the water in a clean dive, is to experience the shock, not of the new, but of the familiar revisited as though it were strange and wonderful.”
—Anna Quindlen (b. 1952)