By William Feller

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Major alterations during this variation contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching strategies, Markov chains, and the De Moivre-Laplace theorem.

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**Extra resources for An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition)**

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Efron, B. (1975). Defining the curvature of a statistical problem (with application to second order efficiency) (with Discussions). Ann. , 1, 1189 - 1242. K. and Subramanyam, K. (1974). Second-order efficiency of maximum likelihood estimators. , 36, 324-358. Kumon, H. a nd Amari, S. (1983). Geometrical theory of higher-order asymptotics of test , interval estimator and conditional inference. Biometrika, (1983), to appear. Nagaoka, H. and Amari, S. (1983). Differential geometry of smooth families of probability distributions.

I In order to prove theorem 1, we need the two following results: A c~n ~ n~ 9! cz>(y,oL)=¢}=r, Lemma 1. 1 [7] ). f { 't "> 0 : ~ ~ ~Y E An 't { I X I;. i 1= ¢} :: r ~m.. inf -itt F(~flxl>fJ) - ~ 't .... +--I} - r f:: (we assume that 00 if 1= 0 ). ~~~a Proof ~ Lemma 1. d. ft). d. P,fl (H) '> O. with the Jio,p (A) = jU (A n{o ~ Ixl

6, 1815-1842. 2. S. Statistical estimation of density function. - Sankhya. 3, 24-5-254. 3. G. Some problems in the spectral analysis of Gaussian random processes. - Teoriya Veroyatnist. i Mat. Statist. (Theory Probab. And Math. lO, 3-11. 4. J. , John Wiley, 1970. 24 5. F. Bias criteria for the selection of spectral windows. - JEEE Trans. Inform. 5,613-615. 6. A. On oomparative characteristics of the estimates for the spectral densi t y functions of stationary random processes. (Probl. Inform. Transmission), 1982, v.