Central Moments in Metric Spaces
Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ +∞.
The pth central moment of a measure μ on the measurable space (M, B(M)) about a given point x0 in M is defined to be
μ is said to have finite pth central moment if the pth central moment of μ about x0 is finite for some x0 ∈ M.
This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, P) is a probability space and X : Ω → M is a random variable, then the pth central moment of X about x0 ∈ M is defined to be
and X has finite pth central moment if the pth central moment of X about x0 is finite for some x0 ∈ M.
Read more about this topic: Moment (mathematics)
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