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. |