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Definition

If 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)}

(where E {\displaystyle \operatorname {E} } 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}}

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})={}&\operatorname {E} (X_{1})\\u_{2}(X_{1},X_{2})={}&\operatorname {E} (X_{1}X_{2})-\operatorname {E} (X_{1})\operatorname {E} (X_{2})\\u_{3}(X_{1},X_{2},X_{3})={}&\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)={}&\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})\\&-\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})\\&+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})\\&+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}}}

Characterization

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.

See also

• Cumulant

• 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

References

1. 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. http://projecteuclid.org/euclid.cmp/1104114545 ↩