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)
Famous quotes containing the words central, moments and/or spaces:
“The central paradox of motherhood is that while our children become the absolute center of our lives, they must also push us back out in the world.... But motherhood that can narrow our lives can also broaden them. It can make us focus intensely on the moment and invest heavily in the future.”
—Ellen Goodman (20th century)
“No man will ever bring out of that office the reputation which carries him into it. The honeymoon would be as short in that case as in any other, and its moments of ecstasy would be ransomed by years of torment and hatred.”
—Thomas Jefferson (1743–1826)
“Deep down, the US, with its space, its technological refinement, its bluff good conscience, even in those spaces which it opens up for simulation, is the only remaining primitive society.”
—Jean Baudrillard (b. 1929)