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.

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!

  • $\begingroup$ A surprising coincidence is that I was very recently discussing exactly this question in my lecture. I'll try to write some ideas. $\endgroup$
    – zhoraster
    Sep 22, 2017 at 10:39

2 Answers 2


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 $X_0,X_1$.

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.

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.

  • $\begingroup$ Great counterexamples! I think it's also feasible to construct a counterexample for "B+independent increments" by combining the second and third one. Thanks! $\endgroup$
    – Dreamer
    Sep 22, 2017 at 21:59

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.


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