Cfrgtkky

I am always considering non-discrete/non-finite probability spaces $Omega$. For everything that follows feel free to assume $Omega = mathbb{R}^n$.

Say you have 2 random variables $X_1,X_2 :Omega rightarrow mathbb{R}$. Now an event based definition of $X_1$ and $X_2$ being ``independent" is as follows : "$X_1$ and $X_2$ are independent random variables if for all $x,y in mathbb{R}$ we have that $mathbb{P}((X_1 leq x)cap(X_2 leq y)) = mathbb{P}(X_1 leq x)mathbb{P}(X_2 leq y)$" (Am I right?)

• Are there natural examples of pairs of independent random variables whose descriptions can be given as maps $Omega rightarrow mathbb{R}$ ?




• Is the above setup enough to ensure that there exists a joint-distribution of $X_1$ and $X_2$? If yes, how?




• When would one prefer to use a joint-distribution based definition of ``independence" as opposed to such an event based defition and vice-versa?

On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).