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:
“For us necessity is not as of old an image without us, with whom we can do warfare; it is a magic web woven through and through us, like that magnetic system of which modern science speaks, penetrating us with a network subtler than our subtlest nerves, yet bearing in it the central forces of the world.”
—Walter Pater (1839–1894)
“All the strong agonized men
Wear the hard clothes of war,
Try to remember what they are fighting for.
But in dark weeping helpless moments of peace
Women and poets believe and resist forever.”
—Muriel Rukeyser (1913–1980)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)