Monte Carlo Estimation of the Density of the Sum of Dependent Random Variables

We study an unbiased estimator for the density of a sum of random variables that are simulated from a computer model. A numerical study on examples with copula dependence is conducted where the proposed estimator performs favourably in terms of variance compared to other unbiased estimators. We provide applications and extensions to the estimation of marginal densities in Bayesian statistics and to the estimation of the density of sums of random variables under Gaussian copula dependence

Magnetic fluctuations in the classical XY model: the origin of an exponential tail in a complex system

We study the probability density function for the fluctuations of the magnetic order parameter in the low temperature phase of the XY model of finite size. In two-dimensions this system is critical over the whole of the low temperature phase. It is shown analytically and without recourse to the scaling hypothesis that, in this case, the distribution is non-Gaussian and of universal form, independent of both system size and critical exponent $\eta$. An exact expression for the generating function of the distribution is obtained, which is transformed and compared with numerical data from high resolution molecular dynamics and Monte Carlo simulations. The calculation is extended to general dimension and an exponential tail is found in all dimensions less than four, despite the fact that critical fluctuations are limited to D=2. These results are discussed in the light of similar behaviour observed in models of interface growth and for dissipative systems driven into a non-equilibrium steady state.Comment: 32 pages, 13 figures, 1 table. Few changes. To appear in Phys. Rev.