Odd use of terms (“stationary stochastic process”., etc.)

[nomadreid]

I am trying to make sense of a Russian author’s use of terms (I have to translate his article). I have three questions, but please don't think you need to answer all three before answering. Thanks for any insights!

[1] He uses the term “probability density distribution” ρ(ξ) of a stationary (stochastic) process ξ.This sounds to me like a hybrid of “probability distribution” and “probability density function”, and that only one of them should be chosen. Am I wrong? If so, the first one?

[2] when talking about stationary processes, he writes in the original “stationary stochastic process”. Is this redundant –shall I eliminate “stochastic”, or leave it in?

[3] When he is referring to deBroglie waves, he writes “deBroglie matter waves”. Is the “matter” definitely to be left out, or can it be left in for emphasis?

 

[andrewkirk]

(1) A stochastic process $\xi$ is a function whose domain is $D=\Omega\times U$ and range is $S$, which is usually $\mathbb R$ or $\mathbb Z$. $\Omega$ is the sample space, which is the set of all possible outcomes. $U$ is the 'index set', which is usually the non-negative reals or the non-negative integers. The index set $U$ usually represents time, but doesn't have to.

A stochastic process can also be considered as a collection of random variables $\left(\xi_t\right)_{t\in U}$, such that $\xi_t(\omega)=\xi(\omega,t)$ is the value of the process at 'time' $t$ in outcome $\omega$. In fact sometimes 'stochastic process' is defined in this way as a collection of random variables, and its nature as a function is derived from that. The two definitions are equivalent.

The stochastic process $\xi$ is stationary if random variables $\xi_t$ and $\xi_s$ have the same distribution, for all $t,s\in U$.

What the author means by “probability density distribution” $\rho(\xi)$ depends on how they use it. If the stoch process is stationary, I think they probably mean the probability density function (pdf) of $\xi_t$, which is the same for all $t\in U$. If it is not stationary, more examination of how it is used would be needed, eg it could be a function $\eta:U\times S\to[0,1]$ such that $\eta(t,\alpha)=f_{\xi_t}(\alpha)$ where $f_{\xi_t}$ is the pdf of $\xi_t$.

(2) I don't think the stationary in 'stationary stochastic process' is redundant, since one can have a deterministic process that is a function from $U$ to $S$. However it is usual when talking about stochastic processes to omit the stochastic after a while, provided it is clear from the context that it means a stochastic process. Since a stationary deterministic process is just a constant function, and it would be rare to characterise a constant function as a process, it is reasonable to use 'stationary process' to refer to stationary stochastic processes. If you want to do this, it can be made clear in the translation by, when the notion of 'stationary' is introduced, and it says something like 'a stochastic process is called stationary if <insert conditions, like those given above>', just adding the words, either in the text or - if you want the translation to be as literal as possible - as a footnote: 'and this may be referred to as a stationary process'.

(3) I'm not aware of any other sort of de Broglie wave. It's usual to call it either a 'matter wave' or a 'de Broglie wave'. If there isn't any other sort of de Broglie wave (I don't think there is, but I don't know for sure) 'de Broglie matter wave' seems unnecessarily long. But it sounds like this author likes to be very formal, so the decision depends on whether you want to go for concision or to preserve the author's style.

 

[nomadreid]

Thank you very much for your detailed and helpful explanations, andrewkirk. Very much appreciated.