Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

382 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswherec(P') = JdP"C(P',P") =Jde»'Jd£'C(a>,E->w',E'jr')the mean number of secondaries in the collision at P'.2. The quantityf NE(P',P") =c(P')f NE(P\P")is scored irrespective of the number of actual secondaries.B. CONFIDENCE LIMITS FOR SINGLULAR ESTIMATORSBecause of the unbounded variance of the next-event estimator, the classical CentralLimit Theorem does not apply to the empirical mean of several estimates by the next-eventestimator, i.e., the unbiasedness of the empirical mean and its rate of convergence to itsexpectation do not follow from the considerations valid for estimates of finite variance(Chapter 3.III.). Kalos has investigated these questions in connection with the next-eventestimator," and Dubi et al. amplified the considerations to cover other estimators with lesssevere singularities. 15Here, we take a more general approach based on a special case of atheorem on stable attracting probability distributions. 18Theorem 6.5 — Let £,, £ 2, . . . ,£„ D e identically distributed random variables with acommon density function p^(x) defined on the semi-infinite interval ( — x 0,+°°). LetdyyP £(y) = 0and define the functionU(x) = j^dyy 2 P c(y) (6.172)Let L(x) be a slowly varying function, i.e.. let it be such thatlimL(xt)/L(x) = 1If, for large values of x, the function U(x) behaves likeU(x) ~ x 2 -"L(x) (6.173)with 1 < a -s 2 and if there exists a sequence a„ such that for some constant CnL(a n)/a*->C (6.174)with increasing n, then the distribution function of the random variables„ = 2 Uk