Probability and Stochastic Processes

2.3 Existence of Random Variables with Prescribed Distribution. Skorohod Representation of a Random Variable

In the previous section, we have seen that any random variable has a distribution function F, which is called in other classes the cdf. Recall the essential properties of this function from Proposition 2.6 on page 47: right continuity, increasing, taking values between 0 and 1. An obvious question is: given a function F with these properties, can we construct a random variable with the desired distribution?

In fact, yes, we can, and this is the first step in a very important theorem we shall see later in this course: the Skorohod representation theorem. However, recall that a random variable has to have as domain some probability space. It actually is true that we can construct random variables with the prescribed distribution on any space, but recall that the purpose of creating random variables was to have a uniform way of treating probability. It is actually enough to give the Skorohod's construction on the probability space , where is the Lebesgue measure.

On this space, for any ω, define the following random variables:

Note that, in statistics, would be called the ...