independent stochastic processes

Two stochastic processes $\{X(t)\mid t\in T\}$ and$\{Y(t)\mid t\in T\}$ are said to be if for any positive integer , and anysequence $t_1,\mathrm{\dots },t_n\in T$, the random vectors$𝑿:=(X(t_1),\mathrm{\dots },X(t_n))$ and$𝒀:=(Y(t_1),\mathrm{\dots },Y(t_n))$ are independent. This means,for any two $n$-dimensional Borel sets $A,B\subseteq {ℝ}^n$,we have

$$P\left[{𝑿}^{-1}(A)\cap {𝒀}^{-1}(B)\right]=P\left[{𝑿}^{-1}(A)\right]P\left[{𝒀}^{-1}(B)\right].$$