Abstract
In this paper, we generalize a result on approximation of sums by sums with a fixed number of summands in [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003] for independent identically distributed summands to the case of independent nonidentically distributed real summands.
We also show that the interpolating functions sine and cosine in the paths of random variables of a special form of the so-called smart path method, introduced in the Bentkus paper, can be successfully replaced by other interpolating functions to obtain the same estimates with the same constants. The obtained results for sums with a fixed number of summands are used to obtain some approximation estimates for random sums by random sums.
We obtain estimates of the difference |Eh(Sn) − Eh(Zn)| for smooth real functions h and of the difference \( \left|\mathbf{E}{\mathrm{e}}^{\mathrm{i}t{S}_n/{b}_n}-\mathbf{E}{\mathrm{e}}^{\mathrm{i}t{Z}_n/{b}_n}\right| \), where bn > 0, of the characteristic functions of the normalized sums Sn/bn and Zn/bn of independent random variables by the smart path interpolation method.
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Sunklodas, J.K. On Some Approximations for Sums of Independent Random Variables. Lith Math J 62, 218–238 (2022). https://doi.org/10.1007/s10986-022-09560-1
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DOI: https://doi.org/10.1007/s10986-022-09560-1