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:
“If women’s role in life is limited solely to housewife/mother, it clearly ends when she can no longer bear more children and the children she has borne leave home.”
—Betty Friedan (20th century)
“The Declaration [of Independence] was not a protest against government, but against the excess of government. It prescribed the proper role of government, to secure the rights of individuals and to effect their safety and happiness. In modern society, no individual can do this alone. So government is not a necessary evil but a necessary good.”
—Gerald R. Ford (b. 1913)
“One of the annoying things about believing in free will and individual responsibility is the difficulty of finding somebody to blame your problems on. And when you do find somebody, it’s remarkable how often his picture turns up on your driver’s license.”
—P.J. (Patrick Jake)
“Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (1889–1968)
“The theory seems to be that so long as a man is a failure he is one of God’s chillun, but that as soon as he has any luck he owes it to the Devil.”
—H.L. (Henry Lewis)