[Math] Probability on product spaces

probabilityprobability theory

I am having some trouble, more of an argument with someone else, about a simple question regarding product spaces.

Let $X_1,X_2,\dots,X_n$ a set of independent and identically distributed random variables from a population $P\in\mathcal{P}$, where $\mathcal{P}$ is a family of probability measures (non explicitly parameterized). The generic random variable $X$ is a measurable function from a probability space $\left(\Omega,\mathcal{F},P'\right)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

In order to build the random vector containing the elements of the sample, $\tilde{X}=[X_i]_{n\times 1}$, and be able to calculate probability measures of events like $\{X_1<X_2\}$ or $\{X_1=X_2\}$, I decided to build a product space $(\Omega^n=\Omega\times\Omega\dots\times\Omega,\sigma(\mathcal{F}^ n)=\sigma(\mathcal{F}\times\mathcal{F} \dots,\times\mathcal{F}),P=P'\times P'\times\dots\times P')$, and let the vector function $\tilde{X}$ go from this product space to $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$.

Therefore, the events

  1. $\{X_1=X_2\}=\{(w_1,w_2,\dots,w_n)\in\Omega^n:X_1(\omega_1)=X_2(\omega_2)\}$
    and
  2. $\{X_1<X_2\}=\{(w_1,w_2,\dots,w_n)\in\Omega^n:X_1(\omega_1)<X_2(\omega_2)\}$

have a clear meaning.

However, the person I am having the argument with argues that

  1. $\tilde{X}$ goes from the original probability space $\left(\Omega,\mathcal{F},P'\right)$ to $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$, and
  2. If I keep doing it the way of product spaces it is a lot harder to calculate the probabilities I want.

If my friend's argument is true, I am having trouble finding the meaning of, and calculating the probabilities of the events above.

What am I missing?

Best regards,

JM

Best Answer

That's a good question. The point is that if you have a random variable $$ X:(\Omega,\mathscr F,P)\to (\Bbb R,\mathscr B(\Bbb R)) $$ and you need to consider, say, two copies of this variable $X_1$ and $X_2$, a direct way would be to define the vector $\tilde X = (X_1,X_2)$ on a product space $(\Omega,\mathscr F,P)\otimes (\Omega,\mathscr F,P)$. This is intuitive, this way always works, and IMHO it's easier to compute the probabilities you've mentioned over a product space - you have clear image of the diagonal in your mind when dealing with $\{X_1 = X_2\}$ and of a subdiagonal triangle when dealing with $\{X_2\leq X_1\}$. The latter make easier computations of the correspondent double integrals.

On the other hand, formally speaking, you don't have to construct a product space in most of the practial cases. That is, most of the probability spaces we're dealing with are standard. For example, since $X$ is a real-valued random variable, you can always take $$ (\Omega,\mathscr F,P) = ([0,1],\mathscr B([0,1]),\lambda) $$ where $\lambda$ is the Lebesgue measure. As a result, the product space is isomorphic to the original space and hence any random vector defined over the product space can be defined over the original space. However, I wouldn't suggest going that way due to the following reasons:

  1. It does not always work: if $\Omega$ has just two elements and $\mathscr F$ is its powerset, then you can't defined $\tilde X$ over the original space.

  2. I disagree with your friend that it is easier to compute probabilities when defining $\tilde X$ over the original state space, rather than over the product space.

  3. It is less intuitive, more technically involved and unnecessary.

Please, tell me whether the answer is clear to you.

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