Abstract
One considers the weighted sums\(S(\alpha (\delta )) = \sum\limits_{k = 0}^\infty {\alpha _k (\delta )} X_k\), where Xk, are independent random variables, having no densities, and αk(δ) are nonnegative functions, defined on the interval (0, 1). One investigates the conditions under which S(α(δ)) has a density; one proves a local limit theorem; one studies the form of the distribution in the case when α(δ)=δk and the variables Xk take only a finite number of values.
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Literature cited
G. Brown and W. Moran, “Coin tossing and powers of singular measures,” Math. Proc. Cambridge Philos. Soc.,77, 349–364 (1975).
H. U. Gerber, “The discounted central limit theorem and its Berry-Esseen analogue,” Ann. Math. Statist.,42, No. 1, 389–392 (1971).
C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York (1979).
C.-G. Esseen, “Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law,” Acta Math.,77, 1–125 (1945).
A. Zygmund, Trigonometric Series. Vol. II, Cambridge Univ. Press, Cambridge (1959).
B. Jessen and A. Wintner, “Distribution functions and the Riemann zeta function,” Trans. Am. Math. Soc.,38, 48–88 (1935).
R. Kershner and A. Wintner, “On symmetric Bernoulli convolutions,” Am. J. Math.,57, No. 3, 541–548 (1935).
P. Levy, “Sur les series dont les termes sont des variables eventuelles independantes,” Stud. Math.,3, 119–155 (1931).
E. Lukacs, Characteristic Functions, Hafner, New York (1970).
V. V. Petrov, “A local theorem for the densities of sums of independent random variables,” Teor. Veroyatn. Primen.,1, No. 3, 349–357 (1956).
J. I. Reich, “When do weighted sums of independent random variables have a density—some results and examples,” Ann. Probab.,10, No. 3, 787–798 (1982).
S. Kh. Sirazhdinov and M. U. Gafurov, “A remark on a certain limit theorem,” in: Random Processes and Statistical Inference [in Russian], Fan, Tashkent (1973), pp. 170–172.
F. Hausdorff, “Momentprobleme fur ein endliches Intervall,” Math. Z.,16, 220–248 (1923).
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Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 83–94, 1986.
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Wolf, W., Sasvari, Z. Some remarks on the existence of densities for sums with weights. J Math Sci 38, 2211–2218 (1987). https://doi.org/10.1007/BF01093822
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DOI: https://doi.org/10.1007/BF01093822