Are the Conditions for q Truly Independent?

[nomadreid]

Working through a paper about whose rigor I have my doubts, but I am always glad to be corrected. In the paper I find the following:
"We now investigate the random variable q. There are the following restrictions on q:
1) The variable q must characterize a stochastic process in the test interval
] t_i - τ; t_i + τ [ as τ tends to zero*;
2) the domain of q is the set of real numbers
3) q must be stochastic"
Two questions: First, "random variable" and "stochastic variable" are synonymous, no? So either (3) or the beginning assumption that q is a random variable would appear to be superfluous.
Second, don't (3) and (2) together imply (1)?
In any case, somehow it seems that these three conditions are not completely independent of one another. I would be grateful for any indications to confirm or deny my intuition.

*PS. The original actually said
"1) The variable q must characterize a stochastic process in the test interval
] t_i - τ₀; t_i + τ [ as τ tends to zero"; but I think that the τ₀ is a typo.

 

[mathman]

The wording is confusing. Is q a random variable or a stochastic process (a function with random variable range)?

 

[Stephen Tashi]

Second, don't (3) and (2) together imply (1)?

I'd say no, but the phrase "characterize a stochastic process" isn't defined.

A stochastic process produces random trajectories. As a literal example, tossing a paper air plane across a drafty room might be modeled a stochastic process. Let v(t) be a specific trajectory that is realized by a stochastic process. Then there is nothing stochastic about v(t). It is a definite trajectory. It's properties, such as $lim_{t\rightarrow 2} v(t)$ are not stochastic. However, we can define a random variable $Q$ by saying: A realization of $Q$ consists of taking a random trajectory v(t) generated by the stochastic process and finding the value of $q = lim_{t\rightarrow 2} v(t)$. This makes $Q$ a random variable by virtue of the fact that $v(t)$ is chosen at random.

If we only say that $Q$ is a random variable and that the domain of $Q$ is the set of real number then this does not imply the structure that $Q$ was defined a function of a random trajectories.

 

[nomadreid]

Thank you, mathman and Stephen Tashi.

The wording is confusing. Is q a random variable or a stochastic process (a function with random variable range)?

I'd say no, but the phrase "characterize a stochastic process" isn't defined.

Yes, his wording is confusing, and of course a problem is that I have quoted out of context. Apparently what he means is that q(t) =x(t) (x-position, t-time) is a stochastic process, and for a specific value of t = t_i, q(t_i)=q is a random variable, then integrating and differentiating over q.

A stochastic process produces random trajectories. As a literal example, tossing a paper air plane across a drafty room might be modeled a stochastic process. Let v(t) be a specific trajectory that is realized by a stochastic process. Then there is nothing stochastic about v(t). It is a definite trajectory. It's properties, such as limt→2v(t)lim_{t\rightarrow 2} v(t) are not stochastic. However, we can define a random variable QQ by saying: A realization of QQ consists of taking a random trajectory v(t) generated by the stochastic process and finding the value of q=limt→2v(t)q = lim_{t\rightarrow 2} v(t) . This makes QQ a random variable by virtue of the fact that v(t)v(t) is chosen at random.

If we only say that QQ is a random variable and that the domain of QQ is the set of real number then this does not imply the structure that QQ was defined a function of a random trajectories.

Thank you, Stephen Tashi, your explanation and counter-example make it clearer. One question about it though: in your example, v(t) is not a random variable, only QQ; in the conditions that I stated, however, he starts out saying that q is a random variable, and then in (3) saying that it must be stochastic: isn't this redundant? Would it have sufficed to either
(a) eliminate (3), or
(b) keep (3) but not start by saying that q was a random variable?

 

[Stephen Tashi]

in the conditions that I stated, however, he starts out saying that q is a random variable,

In what you quoted, he starts out by saying that q must "characterize a stochastic process". It isn't clear whether that phrase implies q is a random variable.

To interpret what you quoted, it would be necessary to understand the context - which includes the intended audience for the paper. For example, articles about applied science sometimes use the terms "random" or "stochastic" in the common language sense of those words instead of in the mathematical sense of "random variable" or "stochastic process".

 

[nomadreid]

Thanks, Stephen Tashi. The intended audience is theoretical physicists; the author proposes an alternative derivation of Schrödinger's equation for a specific case of a classical (pre-quantum) particle in chaotic motion. He uses the term which is Russian for, literally, "stochastic variable", he investigates whether it is correlated with another "random variable", so I interpreted his usage to mean "random variable" in the mathematical sense.