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A formula of full semi-invariants. (English) Zbl 0877.60016
Boccara, Nino (ed.) et al., Cellular automata and cooperative systems. Proceedings of the NATO Advanced Study Institute held in Les Houches, France, June 22-July 2, 1992. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 396, 135-140 (1993).
The author establishes a formula for the semi-invariant \(S_N\) of some system of random variables \((\xi_i\), \(i=1,2, \dots, N)\) which uses the conditional semi-invariants \(S(T\mid \eta)\) of the system of random variables \((\xi_i\), \(i\in T)\), \(T\subset \{1, 2, \dots, N\}\), induced by the conditional distributions of the system under the condition that the value of the random variable \(\eta\) is fixed: \[ S_N= \sum_{\overline T\in {\mathcal T}} S_l \biggl( S\bigl(T_1 \mid\eta \bigr),\;S\bigl(T_2\mid \eta\bigr), \dots, S\bigl( T_l\mid \eta\bigr) \biggr), \] where the sum is taken over the set \({\mathcal T}\) of all partitions \(\overline T=(T_1,T_2, \dots, T_l)\) of the set \(\{1,2, \dots, N\}\). The author indicates that the formula is useful in various probabilistic problems. For example, in a joint paper by the author and the reviewer [Theor. Math. Phys. 20(1974), 782-790 (1975); translation from Teor. Mat. Fiz. 20, 223-234 (1974; Zbl 0311.60063)] this formula was applied for \(N=3\) to obtain a lower bound for the variances of additive functionals of Gibbs random fields.
For the entire collection see [Zbl 0811.00025].
60E99 Distribution theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory