# Finite dimensional distributions

Let $$(\Omega,\mathcal F,P)$$ be a probability space, and $$(S,\mathcal S)$$ a measurable space. A random variable $$\mathcal F$$-$$\mathcal S$$-measurable function $$\Omega\longrightarrow S$$. A random variable is called

• a real-valued random variable if $$S = \mathbb R$$,
• a $$k$$-variate real-valued random variable if $$S=\mathbb R^k$$, and
• a real-valued stochastic process if $$S = \mathbb R^{[0,\infty)}$$.

Any random variable $$X$$ there exists a measure $$P_X$$, called the distribution of $$X$$, which gives raise to a probability space $$(S,\mathcal S,P_X)$$.

Especially in the context of stocahstic processes the notion of finite-dimensional distributions of a random variable $$X$$ is important. Clearly, in the case $$S=\mathbb R$$ and $$S=\mathbb R^k$$ the notion of distribution and finite-dimensional distributions coincide. But in the case $$S=\mathbb R^{[0,\infty)}$$ I get confused by the way finite-dimensional distributions are understood. Are the finite-dimensional distributions the same as $$P_X$$ restricted to all finite-dimensional subspaces of $$\mathbb R^{[0,\infty)}$$?

When $$S=\mathbb{R}^{[0,\infty)}$$, $$\mathscr{S}$$ is defined to be the $$\sigma$$-field $$\mathscr{B}(\mathbb{R^{[0,\infty)}})$$ generated by all the cylinder sets. The sets in the form $$\{u\in\mathbb{R}^{[0,\infty)}:(u(t_1),\cdots,u(t_n))\in A^{(n)}\}$$, where $$0\le t_1<\cdots and $$A^{(n)}\in \mathscr{B}(\mathbb{R}^n)$$, are called cylinder sets. For a stochastic process $$X$$, the distribution of $$X$$ is a measure $$P_X$$ on $$(\mathbb{R}^{[0,\infty)},\mathscr{B}(\mathbb{R^{[0,\infty)}}))$$, defined as $$P_X(A)=\mathbb{P}(X\in A)$$ for any $$A\in \mathscr{B}(\mathbb{R^{[0,\infty)}})$$.
For your question, the finite-dimensional distribution is exactly determined by the values $$P_X(A)$$ for cylinder set $$A$$. And these values determine all the values $$P_X(A)$$ for $$A\in \mathscr{B}(\mathbb{R^{[0,\infty)}})$$, i.e. the distribution of $$X$$. More precisely, if there exist two probability measures $$\mathbb{P}, \mathbb{Q}$$ on $$(\mathbb{R}^{[0,\infty)},\mathscr{B}(\mathbb{R^{[0,\infty)}}))$$ such that $$\mathbb{P}(A)=\mathbb{Q}(A)$$ for any cylinder set $$A$$, then $$\mathbb{P}=\mathbb{Q}$$. You can easily prove it using $$\pi$$-$$\lambda$$ lemma, since cylinder sets are closed under finite intersection. In the book of Billingsley, you can find a more general definition called seperating classes.
• "generated by all the cylinder sets. The sets in the form $\{u\in\mathbb R^{[0,\infty)}:(u(t_1),\dots,u(t_n))\in A^{(n)}\}$, where $0\leq t_1<...<t_n$ and $A^{(n)}\in\mathcal B(\mathbb R^n)$" -- here I have problems regarding the notation. Would it be the same if I said that $\{u\in\mathbb R^{[0,\infty)}: \pi_Iu\in A^{I}\}$, where $\pi_Iu = (u(t_1),\dots,u(t_n))$ with $|I| = n$, and $A^I\in\mathcal B(\mathbb R^I)$: Commented Apr 26, 2021 at 23:39