How much difference is there between the independence of two processes $\{X_t\},\{Y_t\}$ and that of two r.v.'s $X_t,Y_t$ for each $t$?

Question

Let $I$ be an index set, $\{X_t\}_{t\in I},\{Y_t\}_{t\in I}$ be two independent stochastic processes, that is,

for any two finite index subsets $\{t_1,t_2,\cdots,t_n\}, \{s_1,s_2,\cdots,s_m\}$, the two random vectors $(X_{t_1},X_{t_2},\cdots,X_{t_n})$ and $(Y_{s_1},Y_{s_2},\cdots,Y_{s_m})$ are independent.

Then it's obvious that, for any $t\in I$, two r.v.'s $X_t$ and $Y_t$ are independent.

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Now we consider the inverse problem. If we know that $X_t, Y_t$ are independent for any $t\in I$, what else is required to guarantee the independence of two processes $\{X_t\}_{t\in I}$ and $\{Y_t\}_{t\in I}$?

Is it enough to provide $\{X_t\}_{t\in I},\{Y_t\}_{t\in I}$ both have independent increments? I guess it's enough but don't know how to prove.

Any suggestions or comments will be appreciated!

Answer 1

Let us first speak of independent values.

For them,

Condition A The values $X_t$ and $Y_t$ are independent for any $t$

is infinitely far away from independence of $X$ and $Y$. For instance, we can take a process $X$ with independent values and permute its values by setting e.g. $Y_t = X_{t+1}$.

A better idea perhaps is to assume

Condition B The values $X_t$ and $Y_s$ are independent for any $t,s$

But this is still not enough for independence. For example, flip a coin twice and set $X_0 = \mathbf{1}_{\text{first flip is H}}$, $X_1 = \mathbf{1}_{\text{second flip is H}}$, $Y_0 = \mathbf{1}_{\text{the results are identical}}$; the values in other points may be set independently in an arbitrary matter. Then $X$ and $Y$ have independent values and $X_s$, $Y_t$ are independent for each $t,s$, but $Y_0$ does depend on $X_0,X_1$.

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Now concerning the independent increments, it is still possible to construct dependent processes satisfying condition A. Clearly, it is enough to give their values at two points. So let $N_1$ and $N_2$ be independent standard Gaussian random variables and set $X_0 = N_1$, $X_0 = N_1 + N_2$, $Y_0 = N_2$, $Y_1 = N_2 - N_1$. Then they satisfy condition A, have independent increments, but obviously dependent.

So far, I was not able to come up with an example of dependent processes with independent increments satisfying condition B, but I'm pretty sure such processes exist.

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But are there any positive statements?

Clearly, if $X$ and $Y$ satisfy condition B and are jointly Gaussian, then they are independent.

Aside from this, it is hard to imagine any simple assumption allowing to derive independence of $X$ and $Y$ from either A or B.

Answer 2

I post here another possible answer, owing to a friend. To avoid the troublesome produced by the abstract index set $I$ without any structure, we set $I:=\mathbb R_+=[0,\infty)$ from now on. And we assume that two processes both start from $0$, i.e., $X_0=Y_0=0$.

Statement A: Two processes$\{X_t\}_{t\in I},\{Y_t\}_{t\in I}$ are independent.

Statement B: $X_t, Y_t$ are independent for any $t\in I$.

Statement C: The joint process $\{(X_t,Y_t)\}_{t\in I}$ have independent and stationary increments.

Now we can make an assertion.

Proposition. Under Statement C, we have A $\Leftrightarrow$ B.

Proof. It's enough to show that B+C $\Rightarrow$ A. Fix an finite increasing index sequence $0=t_0<t_1<t_2<\cdots<t_n<\infty$, what we want to prove is $(X_{t_1},X_{t_2},\cdots,X_{t_n})$ and $(Y_{t_1},Y_{t_2},\cdots,Y_{t_n})$ are independent, or equivalently, $(X_{t_1}-X_{t_0},X_{t_2}-X_{t_1},\cdots,X_{t_n}-X_{t_{n-1}})$ and $(Y_{t_1}-Y_{t_0},Y_{t_2}-Y_{t_1},\cdots,Y_{t_n}-Y_{t_{n-1}})$ are independent. Now we use the induction argument. Suppose that $(X_{t_1}-X_{t_0},\cdots,X_{t_{n-1}}-X_{t_{n-2}})$ and $(Y_{t_1}-Y_{t_0},\cdots,Y_{t_{n-1}}-Y_{t_{n-2}})$ are independent. Since $\{(X_t,Y_t)\}$ have independent increments, we know that $(X_{t_n}-X_{t_{n-1}},Y_{t_n}-Y_{t_{n-1}})$ is independent with $(X_{t_1}-X_{t_0},Y_{t_1}-Y_{t_0},\cdots,X_{t_{n-1}}-X_{t_{n-2}},Y_{t_{n-1}}-Y_{t_{n-2}})$. By virtue of the stationary increments, $(X_{t_n}-X_{t_{n-1}},Y_{t_n}-Y_{t_{n-1}})$ has the same distribution with $(X_{t_n-t_{n-1}},Y_{t_n-t_{n-1}})$, which has independent components by Statement B. As a consequence, $X_{t_n}-X_{t_{n-1}}$ and $Y_{t_n}-Y_{t_{n-1}}$ are independent. Now we summarize that $(X_{t_1}-X_{t_0},\cdots,X_{t_{n-1}}-X_{t_{n-2}})$, $(Y_{t_1}-Y_{t_0},\cdots,Y_{t_{n-1}}-Y_{t_{n-2}})$, $X_{t_n}-X_{t_{n-1}}$, $Y_{t_n}-Y_{t_{n-1}}$ are mutually independent, what we need then follows.