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)
“Such is the role of poetry. It unveils, in the strict sense of the word. It lays bare, under a light which shakes off torpor, the surprising things which surround us and which our senses record mechanically.”
—Jean Cocteau (1889–1963)
“Everyone asks for freedom for himself,
The man free love, the businessman free trade,
The writer and talker free speech and free press.”
—Robert Frost (1874–1963)
“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)