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

• hojoon yang
In summary: Please provide a link to justify this claim.In summary, Gaussian processes and Poisson processes are both stochastic processes, where the element at a particular value of t has the specified distribution. Changing the scale factor doesn't affect that property.
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?

Last edited:
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.

Helolo
mathman said:
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

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)## .

Helolo said:
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)##.

What do you mean?

Helolo said:
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.

berkeman
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
Mark44 said:
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

Helolo said:
We are talking about Gauss process so the functions ##f## of it are linear

Mark44 said:
You just search on google, ok I am done here. The end of reply

Helolo said:
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.

PWiz
Helolo said:
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).

## 1. What is a gaussian process?

A gaussian process is a type of stochastic process in which any finite set of random variables follows a multivariate normal distribution. In other words, the values of a gaussian process at any given time are normally distributed, and the values at different times are correlated with each other.

## 2. How is X(t) related to X(2t)?

If X(t) is a gaussian process, then X(2t) is also a gaussian process. This means that the values of X(2t) at any given time are normally distributed, and the values at different times are correlated with each other.

## 3. What is the difference between X(t) and X(2t)?

The main difference between X(t) and X(2t) is the rate at which they change over time. X(2t) changes at a faster rate, as it is essentially a compressed version of X(t). However, both processes still follow a multivariate normal distribution and have correlated values at different times.

## 4. Are there any limitations to X(2t) being a gaussian process if X(t) is a gaussian process?

As long as X(t) is a gaussian process, there are no limitations to X(2t) also being a gaussian process. However, it is important to note that the process may change if the scaling factor is changed (i.e. X(kt) may not necessarily be a gaussian process if X(t) is).

## 5. How can the properties of X(t) be used to understand X(2t)?

The properties of X(t) can be used to understand X(2t) in terms of their correlation and distribution. For example, if X(t) has a strong positive correlation between values at different times, then X(2t) will also have a strong positive correlation between values at different times. Additionally, the distribution of X(2t) will still be normal, but with a higher variance due to the faster rate of change.

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