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Q: probability theory - mixture of gaussians ( No Answer,   0 Comments ) Question
 Subject: probability theory - mixture of gaussians Category: Science > Math Asked by: eyaler-ga List Price: \$10.00 Posted: 24 Nov 2006 10:18 PST Expires: 24 Dec 2006 10:18 PST Question ID: 785273
 ```if X1, X2 are random variables normally distributed with mean m and variance s^2, then X1+X2 is normally distributed with mean 2m and variance 2s^2. this may be found by looking at the convolution of X1 and X2. the characteristic function (cf) of the convolution is equal to cf(X1)*cf(X2). this is used to find the time scaling of a series X(t). now suppose Y1, Y2 are each a binary mixture of two gaussians with a density: f(x)=alpha*exp(-(x-m)^2/2s1^2)/(s1*sqrt(2Pi))+ +beta*exp(-(x-m)^2/2s2^2)/(s2*sqrt(2Pi)) where alpha+beta=1. now, this is not a stationary distrubution and one may check that it is not closed under convolution. therefore there is NO scaling for m, s1, s2 such that Y1+Y2 is a binary mixture with scaled parameters. is this correct? and if so, how are binary mixtures of gaussians used to model, say, the probability density of daily stock index returns, while such a model would not be scalable for different time aggregations of the data (e.g. weekly returns). please provide references (preferably online) that address the (non) time-scalability of gaussian mixtures.``` Clarification of Question by eyaler-ga on 25 Nov 2006 05:42 PST ```alpha, beta may also be scaled. one may also look at taylor expansions up to and including s^4 even with these two relaxation, scaling does not seem possible```   