Freydoon Shahidi
In theory of probability, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) is an approximation of the empirical process by a Gaussian process constructed on the same probability space. It is named after Hungarian mathematicians János Komlós, Gábor Tusnády, and Péter Major.
Theory
Let be independent uniform (0,1) random variables. Define a uniform empirical distribution function as
Define a uniform empirical process as
The Donsker theorem (1952) shows that converges in law to a Brownian bridge Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.
- Theorem (KMT, 1975) On a suitable probability space for independent uniform (0,1) r.v. the empirical process can be approximated by a sequence of Brownian bridges such that
- for all positive integers n and all , where a, b, and c are positive constants.
Corollary
A corollary of that theorem is that for any real iid r.v. with cdf it is possible to construct a probability space where independentTemplate:Clarify sequences of empirical processes and Gaussian processes exist such that
References
- Komlos, J., Major, P. and Tusnady, G. (1975) An approximation of partial sums of independent rv’s and the sample df. I, Wahrsch verw Gebiete/Probability Theory and Related Fields, 32, 111–131. Electronic Instrument Positions Staff (Standard ) Cameron from Clarence Creek, usually spends time with hobbies and interests which include knotting, property developers in singapore apartment For sale and boomerangs. Has enrolled in a world contiki journey. Is extremely thrilled specifically about visiting .
- Komlos, J., Major, P. and Tusnady, G. (1976) An approximation of partial sums of independent rv’s and the sample df. II, Wahrsch verw Gebiete/Probability Theory and Related Fields, 34, 33–58. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.