Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Ursell function
open-in-new
<h2 id="definition">Definition</h2> <p>If <i>X</i> is a random variable, the <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> <i>s</i><i>n</i> and cumulants (same as the Ursell functions) <i>u</i><i>n</i> are functions of <i>X</i> related by the <a href="/facts/Exponential_formula/tF9K4Rj9">exponential formula</a>: </p> 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)} <p>(where E {\displaystyle \operatorname {E} } is the <a href="/facts/Expected_value/1XV0JKL8">expectation</a>). </p><p>The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> 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}} <p>The Ursell functions of a single random variable <i>X</i> are obtained from these by setting <i>X</i> = <i>X</i>1 = … = <i>X</i><i>n</i>. </p><p>The first few are given by </p> 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}}} <h2 id="characterization">Characterization</h2> <p>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 <i>X</i><i>i</i> can be divided into two nonempty independent sets. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cumulant/crLC9QLE">Cumulant</a></li></ul> <ul><li><a href="/facts/James_Glimm/gbAJ9bqP">Glimm, James</a>; <a href="/facts/Arthur_Jaffe/0pF6APjR">Jaffe, Arthur</a> (1987), <i>Quantum physics</i> (2nd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-96476-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0887102">0887102</a></li> <li>Percus, J. K. (1975), <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-40/issue-3/Correlation-inequalities-for-Ising-spin-lattices/cmp/1103860532.pdf">"Correlation inequalities for Ising spin lattices"</a> (PDF), <i>Comm. Math. Phys.</i>, 40 (3): 283–308, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1975CMaPh..40..283P">1975CMaPh..40..283P</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01610004">10.1007/bf01610004</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0378683">0378683</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120940116">120940116</a></li> <li>Ursell, H. D. (1927), "The evaluation of Gibbs phase-integral for imperfect gases", <i>Proc. Cambridge Philos. Soc.</i>, 23 (6): 685–697, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1927PCPS...23..685U">1927PCPS...23..685U</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0305004100011191">10.1017/S0305004100011191</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123023251">123023251</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="http://projecteuclid.org/euclid.cmp/1104114545" target="_blank">http://projecteuclid.org/euclid.cmp/1104114545</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>