CTRP를 Fully Characterize하는 방법

🏷️Kolmogorov's Extension/Consistency Theorem

• Method 1: Provide a probability space $(S, A, P)$ and a mapping rule $X(t, s)$
• Method 2: Provide all possible joint CDF/PDF/CF of any finite number of random variables.
• Theorem (Kolmogorov's Extension/Consistency Theorem):
  A continuous-time random process (CTRP) is fully characterized by providing a joint CDF
  $F_{X(t_1), X(t_2), \ldots, X(t_N)}(x_1, x_2, \ldots, x_N)$
  for every $N$-tuple $(t_1, t_2, \ldots, t_N) \in \mathbb{R}^N$ for every $N \in \mathbb{N}$.

1. $F_{X(t)}(x)$ for every $t \in \mathbb{R}$,
2. $F_{X(t_1), X(t_2)}(x_1, x_2)$ for every pair $(t_1, t_2) \in \mathbb{R}^2$,
3. $F_{X(t_1), X(t_2), X(t_3)}(x_1, x_2, x_3)$ for every triplet $(t_1, t_2, t_3) \in \mathbb{R}^3$,
   and so on.

• These are all consistent after any marginalization.
  • 즉, $N$th-order joint distribution에서 magrinalize할 때, 전단계($N-1$th order joint distribution)가 나와야 한다.
• This method works even though there are uncountably infinite jointly distributed random variables.
• Only the joint CDFs of any finite number of random variables are provided.
• Similarly, a discrete-time (DT) random process can be fully characterized.