Analysis of series with stochastic coefficients

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Abstract
For functions with power-law singularities we consider series expansions whose coefficients have been determined by Monte Carlo simulation. In many practical problems the relative noise of the coefficients is constant or slowly increasing with the order. We modeled real Monte Carlo expansions by test series with known singularity structure where noise with different strength and form is imposed on the coefficients. The efficiency of different standard methods of series analysis (ratio method, Padé approximants, differential approximants) has been tested together with smoothing methods based on repeated partial summation of the series. We found the Padé method to give reasonable estimates and its accuracy is independent of the smoothing, while the estimates of the ratio and the differential approximant methods are greatly improved when smoothing is applied. Indeed, we found the ratio method with optimally selected smoothing to give the most reliable results.