[Math] How to understand marginal distribution

measure-theoryprobability theory

  1. Given a random vector $X: (\Omega,
    \mathbb{F}, P) \rightarrow (\prod_{i
    \in I} S_i, \prod_{i \in I}
    \mathbb{S}_i)$, is each component
    variable $X_i, \forall i \in I$ of
    the random vector $X$ always a
    random variable from $(\Omega,
    \mathbb{F}, P)$ to $(S_i,
    \mathbb{S}_i)$?

  2. If yes, I guess there are two ways
    to define the distributions for each
    component variable $X_i, i \in I$:

    • First the random vector $X$ induces a probability measure
      $P_X$ from the its domain to its
      codomain. Then define $P_{X_i}
      (A): = P_{X}(A \times \prod_{j
      \in I, j \neq i} S_j), \forall A
      \in \mathbb{S}_i$.
    • $X_i$ can induce a probability measure $P'_{X_i}$ from its domain
      to its codmain.

    I was wondering if $P_{X_i}$ and
    $P'_{X_i}$ are always the same on
    $\mathbb{S}_i$?

    Will the first definition of marginal probability measures make the product of all the marginal probability measures to be the same as the joint measure?

  3. Can the above definitions and their
    relation be generalized to arbitrary
    measure space $(\Omega, \mathbb{F},
    \mu)$ and mearuable mapping
    $f:(\Omega, \mathbb{F}) \rightarrow
    (\prod_{i \in I} S_i, \prod_{i \in
    I} \mathbb{S}_i)$, to define the
    distribution of each component
    mapping $f_i $ of $f$?

    Does the first way of definition
    require that the measure $\mu$ must
    be a probability measure? How to
    adjust the definition when $\mu$ can
    be any measure? Shall one define $\mu_{f_i}$ from $\mu_f$ as
    $\mu_{f_i}(A_i):=
    \frac{\mu_f(A_i \times \prod_{j \in
    I, j\neq i} S_i)}{\prod_{j \in I,
    j\neq i} \mu_{f_j}(S_i)}$, or $\mu_{f_i}(A_i):=
    \mu_f(A_i \times \prod_{j \in
    I, j\neq i} S_i)$, $\forall A_i \in \mathbb{S}_i $ or something
    else?

    When $\mu$ can be any measure, if it
    is not possible to define in the
    first way, or it is possible but the
    two definitions are not the same,
    how about when $\mu$ is finite, i.e.
    $\mu(\Omega) < \infty$?

Thanks and regards!

Best Answer

First, $\prod_{i\in I}\mathbb S_i$ should be understood as the product $\sigma$-algebra, i.e., the smallest $\sigma$-algebra on $\prod_{i\in I}S_i$ such that the projection $\pi_i:\prod_{j\in I}S_j \ni (s_j)_{j\in I}\mapsto s_i\in \mathbb S_i$ is measurable. In this case, then the answers to the first two questions are Yes.

For part (3), as explained in the comments and answers in another closely related post Measure from on product $\sigma$-algebra to on component $\sigma$-algebras, one needs to work with finite measures on the product space. In this case, without loss of generality assuming $\mu = \mathbb P$ is a probability measure and using $X$ instead of $f$, one recovers part (2).

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