By V. M. Kalinin
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Extra info for Investigations in Classical Problems of Probability Theory and Mathematical Statistics: Part I
S) When we have l:. 16) by parts for V=-2 and 1:. ). +t)dtl. +~)lM,l~+~)o Rem ark 2. 7) with 'i: 2, ~, .. 6) transforms into the well-known Weierstrass infinite product for the gamma function. o In particular, this suggests yet another method for proving Theorem 9: Weierstrass' formula provides the starting point for induction, while an integration by parts with the Bernoulli polynomials introduced into the integrand completes it. Remark 3. ). 19) only, since the natural variable in applications is usually 'X.
If property (0\2) of the Bernoulli polynomials is used. d R em ark. 83) can, of course, be written O(~). J~+e. KALININ 54 for 'l:. ~o , s ~-~) andl4~-~. 88) By Theorem 14, we can give the following representation for the function 00 Q. Att • . +i)'t. 'f, (~+-\;)t. o (U-V ... l) ( ~ .. e) IH\ (h.. e »;1] U" p-2~H ,. ~ •.... Here and in the first two integrals we have ~"t1 - ~t - Q,. 89) should be replaced by 1. e 0 roll a r y. _00. 90) where 9-a. , we can also represent the gamma function as an infinite product with a factor of eIlf(f)Theor em 15.
It is not difficult to estimate the order of magnitude of the remainder term. It only remains now to rearrange the double sum in powers of Rem ark. 71). An exact expression can also be obtained here for the remainder term, although it is cumbersome. Theorem 17. For 'l:,l>O, f=1,2 ... ,3, .... Proof. 99) and the simplification of the resulting formula. This theorem will be found to be very useful together with Theorem 16 for the derivation of probabilistic limit theorems with large deviations. Remark 1.