Abstract
Wald and a number of other American authors have given interesting theorems concerning the sums of the first v random variables from an infinite sequence where the number v of terms is a random variable (see [1]–[3], where references to earlier literature can be found). In their method of proof these theorems go back to the work of one of the authors of the present paper [4], where for estimating the probability he considered sums ζ v with index v equal to the first number n for which
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Uspekhi Mat. Nauk4:4 (1949), 168-172.
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References
A. Wald, ‘On cumulative sums of random variables’, Ann. Math. Statist. 15 (1944), 283–296.
A. Wald, ‘Differentiation under the expectation sign of the fundamental identity in sequential analysis’, Ann. Math. Statist. 17 (1946).
J. Wolfowitz, ‘The efficiency of sequential estimates and Wald’s equation for sequential processes’, Ann. Math. Statist. 18 (1947), 215–231.
A.N. Kolmogoroff, ‘Über die Summen durch den Zufall bestimmter unabhängiger Grossen’, Math. Ann. 99 (1928), 309–319.
S.N. Bernshtein, Probability theory, GTTI, Moscow-Leningrad, 1946 (in Russian).
A.N. Kolmogorov, Fundamentals of probability theory, ONTI, Moscow-Leningrad, 1936 (in Russian).
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Prokhorov, Y.V. (1992). On Sums of A Random Number of Random Terms. In: Shiryayev, A.N. (eds) Selected Works of A. N. Kolmogorov. Mathematics and Its Applications (Soviet Series), vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2260-3_35
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DOI: https://doi.org/10.1007/978-94-011-2260-3_35
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