Role in Free Probability Theory
The lattice of noncrossing partitions plays the same role in defining "free cumulants" in free probability theory that is played by the lattice of all partitions in defining joint cumulants in classical probability theory. To be more precise, let be a non-commutative probability space, a non-commutative random variable with free cumulants . (See free probability for terminology.) Then
where denotes the number of blocks of length in the non-crossing partition . That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. This is the free analogue of the moment-cumulant formula in classical probability. See also Wigner semicircle distribution.
Read more about this topic: Noncrossing Partition
Famous quotes containing the words role, free, probability and/or theory:
“Of course, some men are very effective caregivers [of elderly parents]. But this situation occurs far less frequently for males than females, because it is a role reversal. For women, caregiving is an expected duty; for men, it is an unexpected expression of love or devotion.”
—Tish Sommers (20th century)
“It is only when we speak what is right that we stand a chance at night of being blown to bits in our homes. Can we call this a free country, when I am afraid to go to sleep in my own home in Mississippi?... I might not live two hours after I get back home, but I want to be a part of setting the Negro free in Mississippi.”
—Fannie Lou Hamer (1917–1977)
“Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical probability provides loopholes, and discrepancies are expunged by Faith.”
—Ursula K. Le Guin (b. 1929)
“Freud was a hero. He descended to the “Underworld” and met there stark terrors. He carried with him his theory as a Medusa’s head which turned these terrors to stone.”
—R.D. (Ronald David)