About stochastic process....Help please

[hojoon yang]

Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes.

(a)X(2t)

solution said that X(2t) is not gaussian process, since

and similarly

Given Poisson process X(t)

(a) X(2t)

soultion said that X(2t) is not poisson process, since same reason above.

BUT

I think that in stochastic process, Time t is just constant value.

so I think X(2t), X(10000t), X(t+100) is also gaussian process ,or poisson process

doesn't care about whatever t is.

answer is?

 

[Ray Vickson]

Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes.

(a)X(2t)

solution said that X(2t) is not gaussian process, since

View attachment 84875

and similarly

Given Poisson process X(t)

(a) X(2t)

soultion said that X(2t) is not poisson process, since same reason above.

View attachment 84876

BUT

I think that in stochastic process, Time t is just constant value.

so I think X(2t), X(10000t), X(t+100) is also gaussian process ,or poisson process

doesn't care about whatever t is.

answer is?

You are correct; if one looks at the usual definition of a Gaussian process, Y(t) =X(2t) satisfies the definition. However, its $\mu$ and $\sigma$ are different from those of X(t). Maybe your book uses some really weird definition of Gaussian process, but I hope not---as that would be misleading generations of students. See, eg., https://en.wikipedia.org/wiki/Gaussian_process . The same remarks apply to your Poisson process case.

Frankly, I am surprised someone would make those types of errors, because the scaling properties (of Poisson processes, in particular) are absolutely fundamental in modelling and applications.

 

[hojoon yang]

You are correct; if one looks at the usual definition of a Gaussian process, Y(t) =X(2t) satisfies the definition. However, its $\mu$ and $\sigma$ are different from those of X(t). Maybe your book uses some really weird definition of Gaussian process, but I hope not---as that would be misleading generations of students. See, eg., https://en.wikipedia.org/wiki/Gaussian_process . The same remarks apply to your Poisson process case.

Frankly, I am surprised someone would make those types of errors, because the scaling properties (of Poisson processes, in particular) are absolutely fundamental in modelling and applications.

Thanks for reply vickson!