Science > Physics > Stochastic Processes: Gaussian, Second-Order, etc.
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
28 Aug 2005 02:18:56 PM |
| Object: |
Stochastic Processes: Gaussian, Second-Order, etc. |
From Osher Doctorow
COPYRIGHT NOTICE
Stochastic Processes: Gaussian, Second-Order, etc.
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
A stochastic process is any collection of random variables {X(t)} with
t in some real-valued index set, often taken to be time.
A Second Order (stochastic being understood) Process has finite second
moment for each t:
1) EX^2(t) < infinity, all t in index set
A Gaussian Process has every finite linear combination of random
variables normally/Gaussian distributed.
The Wiener Process {W(t)} is a model for Brownian Motion and has
independent normally distributed increments (diffeences of random
variables) as follows:
2) W(t) - W(s) is normal/Gaussian w mean 0 and variance o^2(t - s), s <
= t
3) W(ti) - W(ti-1) are independent for t1 < = t2 < = ... < = tn, i = 2
to n.
W(0) is often taken as 0, and o^2 is a positive constant.
The Poisson Process {P(t)} is exactly like (2) and (3) above except
that "normal/Gaussian" is replaced by Poisson, which is a discrete
distribution (although with countably many values) and o^2 is replaced
by Lambda (a constant) for 0 < = s < = t.
The Levy process is like Gaussian and Wiener and Poisson processes but
with any infinitely divisible distribution replacing these particular
types. I'll define infinitely divisible distribution later hopefully.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Stochastic Processes: Gaussian, Second-Order, etc. |
28 Aug 2005 02:48:35 PM |
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From Osher Doctorow
An infinitely divisible distribution is such that its characteristic
function has an nth root which is also a characteristic function for
all positive integer n, where E(e^(itX) is the characteristic function
phi_X(t) of random variable X, t any real number.
Levy Processes are also called Levy Stable or just Stable Processes and
other than in special cases like Brownian Motion have infinite variance
Fractional Brownian Motion Processes (fBM or FBM) has normally
distributed increments but they're not independent.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Stochastic Processes: Gaussian, Second-Order, etc. |
28 Aug 2005 02:56:47 PM |
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From Osher Doctorow
Cauchy, Poisson, and normal/Gaussian distributions are infinitely
divisible distributions.
The various types of stochastic processes listed in this thread are
useful for my other recent threads including the last two threads.
Osher Doctorow
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