Why care distribution functions more than random variables?

Question

This may be wrong, but I have often heard some saying " we mainly care about CDFs". Similarly, in textbooks, one sees $X \sim N(0,1)$, without any reference to sample space. But why - and how do we justify this?

My thoughts: with my limited knowledge in probability, two results:

• Any distribution function $F:\mathbb{R} \rightarrow [0,1]$, yields a Lebesgue-Stiltjes measure $\mu_F$. Considering the space $(\mathbb{R}, B_{\mathbb{R}}, \mu_F)$ with randon variable $X$ being identity, we obtain, $$P(X \le t ) = \mu((-\infty, t]) = F(t)$$

This implies it is sufficient in specifying cdf $F$, and say there exists a RV, $X$ with cdf $F$.

• (Skorokhod's construction, in Williams) Let $F: \mathbb{R} \rightarrow [0,1]$ be a cdf. $U\sim U[0,1]$. $$ X^-:= \sup \{ y \in \mathbb{R} \,: \, F(y) < U \} $$ is a RV on $[0,1]$ with same distribution as $F$.

I believe there is also a generalization to joint variables.

But these results are not satisfying, I don't see how they are canonical.

Answer 1

Because of the Kolmogorov existence theorem. It tells us that given any distributions (not defined in a dumb way), there exists a stochastic process (in particular, a random variable) with those distributions.

Often we do not care about the actual probability space. The entire point of a random variable is to assign numbers to outcomes. Real numbers are easier to deal with than generic probability spaces. Although there are times where the space itself is important.

Taken from Wikipedia:

Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.

The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process. The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.

The theorem is used in one of the standard proofs of existence of a Brownian motion, by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above. As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, and one then uses the Kolmogorov continuity theorem to construct a continuous modification of the process constructed by the Kolmogorov extension theorem.