The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.[1]
The Ursell functions of a single random variable X are obtained from these by setting X = X1 = … = Xn.
The first few are given by
Characterizationedit
Percus (1975) showed that the Ursell functions, considered as multilinear functions of several random variables, are uniquely determined up to a constant by the fact that they vanish whenever the variables Xi can be divided into two nonempty independent sets.
^Shlosman, S. B. (1986). "Signs of the Ising model Ursell functions". Communications in Mathematical Physics. 102 (4): 679–686. Bibcode:1985CMaPh.102..679S. doi:10.1007/BF01221652. S2CID 122963530.
Percus, J. K. (1975), "Correlation inequalities for Ising spin lattices" (PDF), Comm. Math. Phys., 40 (3): 283–308, Bibcode:1975CMaPh..40..283P, doi:10.1007/bf01610004, MR 0378683, S2CID 120940116
Ursell, H. D. (1927), "The evaluation of Gibbs phase-integral for imperfect gases", Proc. Cambridge Philos. Soc., 23 (6): 685–697, Bibcode:1927PCPS...23..685U, doi:10.1017/S0305004100011191, S2CID 123023251
January 01, 1970
ursell, function, statistical, mechanics, connected, correlation, function, cumulant, random, variable, often, obtained, summing, over, connected, feynman, diagrams, over, feynman, diagrams, gives, correlation, functions, named, after, harold, ursell, introduc. In statistical mechanics an Ursell function or connected correlation function is a cumulant of a random variable It can often be obtained by summing over connected Feynman diagrams the sum over all Feynman diagrams gives the correlation functions The Ursell function was named after Harold Ursell who introduced it in 1927 Contents 1 Definition 2 Characterization 3 See also 4 ReferencesDefinition editIf X is a random variable the moments sn and cumulants same as the Ursell functions un are functions of X related by the exponential formula E exp z X n s n z n n exp n u n z n n displaystyle operatorname E exp zX sum n s n frac z n n exp left sum n u n frac z n n right nbsp where E displaystyle operatorname E nbsp is the expectation The Ursell functions for multivariate random variables are defined analogously to the above and in the same way as multivariate cumulants 1 u n X 1 X n z 1 z n log E exp z i X i z i 0 displaystyle u n left X 1 ldots X n right left frac partial partial z 1 cdots frac partial partial z n log operatorname E left exp sum z i X i right right z i 0 nbsp The Ursell functions of a single random variable X are obtained from these by setting X X1 Xn The first few are given by u 1 X 1 E X 1 u 2 X 1 X 2 E X 1 X 2 E X 1 E X 2 u 3 X 1 X 2 X 3 E X 1 X 2 X 3 E X 1 E X 2 X 3 E X 2 E X 3 X 1 E X 3 E X 1 X 2 2 E X 1 E X 2 E X 3 u 4 X 1 X 2 X 3 X 4 E X 1 X 2 X 3 X 4 E X 1 E X 2 X 3 X 4 E X 2 E X 1 X 3 X 4 E X 3 E X 1 X 2 X 4 E X 4 E X 1 X 2 X 3 E X 1 X 2 E X 3 X 4 E X 1 X 3 E X 2 X 4 E X 1 X 4 E X 2 X 3 2 E X 1 X 2 E X 3 E X 4 2 E X 1 X 3 E X 2 E X 4 2 E X 1 X 4 E X 2 E X 3 2 E X 2 X 3 E X 1 E X 4 2 E X 2 X 4 E X 1 E X 3 2 E X 3 X 4 E X 1 E X 2 6 E X 1 E X 2 E X 3 E X 4 displaystyle begin aligned u 1 X 1 amp operatorname E X 1 u 2 X 1 X 2 amp operatorname E X 1 X 2 operatorname E X 1 operatorname E X 2 u 3 X 1 X 2 X 3 amp operatorname E X 1 X 2 X 3 operatorname E X 1 operatorname E X 2 X 3 operatorname E X 2 operatorname E X 3 X 1 operatorname E X 3 operatorname E X 1 X 2 2 operatorname E X 1 operatorname E X 2 operatorname E X 3 u 4 left X 1 X 2 X 3 X 4 right amp operatorname E X 1 X 2 X 3 X 4 operatorname E X 1 operatorname E X 2 X 3 X 4 operatorname E X 2 operatorname E X 1 X 3 X 4 operatorname E X 3 operatorname E X 1 X 2 X 4 operatorname E X 4 operatorname E X 1 X 2 X 3 amp operatorname E X 1 X 2 operatorname E X 3 X 4 operatorname E X 1 X 3 operatorname E X 2 X 4 operatorname E X 1 X 4 operatorname E X 2 X 3 amp 2 operatorname E X 1 X 2 operatorname E X 3 operatorname E X 4 2 operatorname E X 1 X 3 operatorname E X 2 operatorname E X 4 2 operatorname E X 1 X 4 operatorname E X 2 operatorname E X 3 2 operatorname E X 2 X 3 operatorname E X 1 operatorname E X 4 amp 2 operatorname E X 2 X 4 operatorname E X 1 operatorname E X 3 2 operatorname E X 3 X 4 operatorname E X 1 operatorname E X 2 6 operatorname E X 1 operatorname E X 2 operatorname E X 3 operatorname E X 4 end aligned nbsp Characterization editPercus 1975 showed that the Ursell functions considered as multilinear functions of several random variables are uniquely determined up to a constant by the fact that they vanish whenever the variables Xi can be divided into two nonempty independent sets See also editCumulantReferences edit Shlosman S B 1986 Signs of the Ising model Ursell functions Communications in Mathematical Physics 102 4 679 686 Bibcode 1985CMaPh 102 679S doi 10 1007 BF01221652 S2CID 122963530 Glimm James Jaffe Arthur 1987 Quantum physics 2nd ed Berlin New York Springer Verlag ISBN 978 0 387 96476 8 MR 0887102 Percus J K 1975 Correlation inequalities for Ising spin lattices PDF Comm Math Phys 40 3 283 308 Bibcode 1975CMaPh 40 283P doi 10 1007 bf01610004 MR 0378683 S2CID 120940116 Ursell H D 1927 The evaluation of Gibbs phase integral for imperfect gases Proc Cambridge Philos Soc 23 6 685 697 Bibcode 1927PCPS 23 685U doi 10 1017 S0305004100011191 S2CID 123023251 Retrieved from https en wikipedia org w index php title Ursell function amp oldid 1123363686, wikipedia, wiki, book, books, library,
