Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

384 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThis statement is meaningful only if lim a n/n =0. Now if the random variable inquestion satisfies the conditions of Theorem 6.5, then from Equation (6.174),. Ua n) .. a„lim -—- ; : lim —with 0 < a =£ ). Thus, it remains to show that for a slowly varying function L(x), thelimiting relationlim L(XVx 1 * = 0holds for a > 0. Assume that the opposite statement holds, i.e., suppose that, for somenonzero Alim L(x)/x a = A 7 0Then obviouslylim L(tx)/(tx)" =Afor any t > 0, and therefore.. L(tx)/(tx)°urn ——— = 1L(x)/x"On the other hand, since L(x) is slowly varyingL(tx) (tx)« (tx)°t a^ ,km ~ — = lim —— == t" ^ 1x—->co L(^x) x xwhich contradiction is resolved if and only if A = 0, i.e. if a n/n tends to zero with increasingn.It is remarkable that the confidence limits are expressed in terms of the expected value,R, in contrast to the nonsingular case where the standard deviation appears in the confidencelimits.It remains to determine the values of a„. Let us consider the general case when thesingularity of the final score in a history has the form(x(r) ~ R/[(k + I)r k ]where r = |r — r*|, the minimum distance of the collision points in the trajectory from thedetector point. [We leave the exponent k undefined instead of setting k = 2 because of theestimators with a singularity 1/r (k = 1) to be investigated in the next Section.] For thesake of simplicity, we assume that the probability density function of the minimum distancer isp