Course in Probability Theory

7 .6 INFINITE DIVISIBILITY 1 261ch .f., the convergence is uniform in every finite interval by the convergencetheorem for ch .f .'s . Alternatively, ifco(t) = lim e`n` tn-+a0then ~o satisfies Cauchy's functional equation and must be of the form e"t,which is a ch .f. These approaches are fancier than the simple one indicatedin the hint for the said exercise, but they are interesting . There is no knownquick proof by "taking logarithms", as some authors have done .]Bibliographical NoteThe most comprehensive treatment of the material in Secs . 7 .1, 7 .2, and 7 .6 is byGnedenko and Kolmogorov [12] . In this as well as nearly all other existing bookson the subject, the handling of logarithms must be strengthened by the discussion inSec . 7 .6 .For an extensive development of Lindeberg's method (the operator approach) toinfinitely divisible laws, see Feller [13, vol . 2] .Theorem 7 .3 .2 together with its proof as given here is implicit inW . Doeblin, Sur deux problemes de M. Kolmogoroff concernant les chainesdenombrables, Bull . Soc . Math . France 66 (1938), 210-220 .It was rediscovered by F . J. Anscombe . For the extension to the case where the constantc in (3) of Sec . 7 .3 is replaced by an arbitrary r.v ., see H . Wittenberg, Limiting distributionsof random sums of independent random variables, Z . Wahrscheinlichkeitstheorie1 (1964), 7-18 .Theorem 7 .3 .3 is contained inP . Erdos and M . Kac, On certain limit theorems of the theory of probability, Bull .Am . Math . Soc . 52 (1946) 292-302 .The proof given for Theorem 7 .4 .1 is based onP . L . Hsu, The approximate distributions of the mean and variance of a sample ofindependent variables, Ann . Math . Statistics 16 (1945), 1-29 .This paper contains historical references as well as further extensions .For the law of the iterated logarithm, see the classicA . N . Kolmogorov, Uber das Gesetz des iterierten Logarithmus, Math . Annalen101 (1929), 126-136 .For a survey of "classical limit theorems" up to 1945, seeW . Feller, The fundamental limit theorems in probability, Bull . Am . Math . Soc .51 (1945), 800-832 .