If X(t) is gaussian process, How about X(2t)?

[hojoon yang]

written as title,

1.
If X(t) is gaussian process, then

Can I say that X(2t) is gaussian process?

of course, 2*X(t) is gaussian process

2. If X(t) is poisson process, then

X(2t) is also poisson process?

 

[mathman]

In both cases you have a stochastic process where the element at a particular value of t has the specified distribution. Changing the scale factor doesn't affect that property.

 

[hojoon yang]

In both cases you have a stochastic process where the element at a particular value of t has the specified distribution. Changing the scale factor doesn't affect that property.

THANKS

 

[Helolo]

Of course when $X(t)$ is gauss process, $X(2t)$ is also gauss because $X(2t)=2X(t)$ applied by $f(\alpha x)=\alpha f(x)$ .

 

[Mark44]

Of course when $X(t)$ is gauss process, $X(2t)$ is also gauss because $X(2t)=2X(t)$ applied by $f(\alpha x)=\alpha f(x)$ .

I don't know anything about Gaussian processes, but I doubt very much that X(2t) = 2X(t). For most functions, $f(\alpha x) \neq \alpha f(x)$.

 

[Helolo]

What do you mean?

 

[Mark44]

What do you mean?

What I mean is that, in general, functions aren't linear. Here are a few examples:
$\cos(2x) \neq 2\cos(x)$
$\sqrt{2x} \neq 2\sqrt{x}$
$\ln(2x) \neq 2\ln(x)$
$10^{2x} \neq 2\cdot 10^x$
etc.

 

[FactChecker]

A Gaussian process, X(y), in variable y is just any process that is a normally distributed random variable for every value of y. It is not important the the variable y is time or not. If X(t) is a Gaussian process in time t, let y=2t, The process X(y) is a normally distributed random variable at y whose parameters, mean and variance are the same as X(t) where t=y/2 . So X(y), y=2t is also a Gaussian process. The Poisson process is similar, but not as simple. Since the Poison properties are very dependent on time t, changing to another variable y=2t is not so obvious. You should check if all the Poisson properties still apply with the new variable y.

 

[Helolo]

What I mean is that, in general, functions aren't linear. Here are a few examples:
$\cos(2x) \neq 2\cos(x)$
$\sqrt{2x} \neq 2\sqrt{x}$
$\ln(2x) \neq 2\ln(x)$
$10^{2x} \neq 2\cdot 10^x$
etc.

We are talking about Gauss process so the functions $f$ of it are linear

 

[Mark44]

We are talking about Gauss process so the functions $f$ of it are linear

Please provide a link to justify this claim.

 

[Helolo]

Please provide a link to justify this claim.

You just search on google, ok I am done here. The end of reply

 

[Mark44]

You just search on google, ok I am done here. The end of reply

You made the claim - it's up to you to justify it.

 

[FactChecker]

Of course when $X(t)$ is gauss process, $X(2t)$ is also gauss because $X(2t)=2X(t)$ applied by $f(\alpha x)=\alpha f(x)$ .

That is not right. Saying a process is Gaussian doesn't say anything about the relationship between X(t) and X(2t).