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 in, role, free, probability and/or theory:
“Always and everywhere children take an active role in the construction and acquisition of learning and understanding. To learn is a satisfying experience, but also, as the psychologist Nelson Goodman tells us, to understand is to experience desire, drama, and conquest.”
—Carolyn Edwards (20th century)
“So successful has been the camera’s role in beautifying the world that photographs, rather than the world, have become the standard of the beautiful.”
—Susan Sontag (b. 1933)
“There the wicked cease from troubling, and there the weary are at rest. There the prisoners are at ease together; they do not hear the voice of the taskmaster. The small and the great are there, and the slaves are free from their masters.”
—Bible: Hebrew, Job 3:17-19.
“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)
“Hygiene is the corruption of medicine by morality. It is impossible to find a hygienest who does not debase his theory of the healthful with a theory of the virtuous.... The true aim of medicine is not to make men virtuous; it is to safeguard and rescue them from the consequences of their vices.”
—H.L. (Henry Lewis)