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Statistical uncertainty estimation of higher-order cumulants with finite efficiency and its application in heavy-ion collisions

We derive the general analytical expressions for the statistical uncertainties of cumulants up to fourth order including an efficiency correction. The analytical expressions have been tested with a toy Monte Carlo model analysis. An application to the study of particle multiplicity fluctuations in heavy-ion collisions is investigated. In this derivation, a mathematical proof is given that the validity of an averaged efficiency correction and the fluctuations induced by the non-uniformity of efficiency can be eliminated. The estimation of statistical uncertainties using the analytical formulas is found to be significantly faster than the commonly used bootstrap method. The simplicity and efficiency of using the analytical formulas may be useful for massive data analysis in many fields.