1

To prove that two processes $(X_t)$ and $(Y_t)$ has the same distribution, I proved that $X_t\sim Y_t$, but my teacher said that it's not enough, I have to prove that they I the same finite dimensional distribution, i.e. that $(X_{t_1},...,X_{t_n})$ and $(Y_{t_1},...,Y_{t_n})$ has same distribution for all $t_0<...<t_n$. So could you give me an example of processes s.t. $X_t\sim Y_t$ for all $t$, but $(X_t)$ and $(Y_t)$ has not the same finite dimensional distribution ? I can't find any.

Math1000
  • 38,041
user657324
  • 1,953
  • What were the processes in particular for which you proved that $X_t\sim Y_t$ for all $t$? – Math1000 Nov 01 '19 at 19:45
  • 2
    For a quick example, let $(Z_n){n\geq 0}$ be a sequence of i.i.d. non-degenerate RVs and set $$X_t=Z{\lfloor t\rfloor}, \qquad Y_t = Z_0. $$ Then $X_t\sim Y_t$ for each $t \geq 0$ but $(X_0, X_1) \not\sim (Y_0, Y_1)$. – Sangchul Lee Nov 01 '19 at 20:35
  • Consider any distribution $\mu,$ and denote $(X_t)_{t \geq 0}$ the process (a family) of i.i.d.r.v. with law $\mu.$ Set $Y_t = X_0$ for all $t.$ – William M. Nov 02 '19 at 02:30

0 Answers0