Independent stochastic processes and independent random vectors

  1. The definition for the two processes
    to be independent is given by
    PlanetMath:

    Two stochastic processes {X(t)tT} and {Y(t)tT} are said to
    be independent, if for any positive
    integer n<, and any sequence
    t1,,tnT, the random
    vectors
    \boldsymbol{X}:=(X(t_1),\ldots,X(t_n))
    and
    \boldsymbol{Y}:=(Y(t_1),\ldots,Y(t_n))
    are independent.

    I was wondering if according to the
    definition, for any positive integer
    n,m<\infty, and any sequence
    t_1,\ldots,t_n\in T and any
    sequence s_1,\ldots,s_m\in T, the
    random vectors
    \boldsymbol{X}:=(X(t_1),\ldots,X(t_n))
    and
    \boldsymbol{Y}:=(Y(s_1),\ldots,Y(s_m))
    are also independent?

  2. Some related questions are for two
    independent random vectors V and
    W:

    • Will any subvector of V and any subvector of W (the two subvectors
      do not necessarily have the same indices in the original random vectors) be independent?
    • For any two subvectors V_1 and V_2 of V and any two subvectors
      W_1 and W_2 of W, will the conditional random vectors V_1|V_2
      and W_1|W_2 also be independent?

Thanks and regards!

Answer

The answer to all your questions is yes. And they can be deduced from the following :

If two random vectors \boldsymbol{X}:=(X_1, \ldots,X_n)
and \boldsymbol{Y}:=(Y_1, \ldots,Y_m) are independent, any pair of "marginalized" random vectors \boldsymbol{X_A}, \boldsymbol{Y_B} (each formed by arbitrary subsets of the originals) are independent.

This property (basically your second question) can be readily deduced from the definition of independence (factorization of joint densities) and marginalization. From this the other two follow.

Attribution
Source : Link , Question Author : Tim , Answer Author : leonbloy

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