In 1945, very early in the history of the development of a rigorous analytical theory of probability, Feller (1945) wrote a paper called “The fundamental limit theorems in probability” in which he set out what he considered to be “the two most important limit theorems in the modern theory of probability: the central limit theorem and the recently discovered … ‘Kolmogoroff’s cel ebrated law of the iterated logarithm’ ”. A little later in the article he added to these, via a charming description, the “little brother (of the central limit theo rem), the weak law of large numbers”, and also the strong law of large num bers, which he considers as a close relative of the law of the iterated logarithm. Feller might well have added to these also the beautiful and highly applicable results of renewal theory, which at the time he himself together with eminent colleagues were vigorously producing. Feller’s introductory remarks include the visionary: “The history of probability shows that our problems must be treated in their greatest generality: only in this way can we hope to discover the most natural tools and to open channels for new progress. This remark leads naturally to that characteristic of our theory which makes it attractive beyond its importance for various applications: a combination of an amazing generality with algebraic precision.
قائمة المحتويات
Commentary: Author’s Pick, by C. C. Heyde.- Commentary: Chris Heyde’s Contribution to Inference in Stochastic Processes, by Ishwar Basawa.- Commentary: Chris Heyde’s Work on Rates of Convergence in the Central Limit Theorem, by Peter Hall.- Commentary: Chris Heyde’s Work in Probability Theory, with an Emphasis on the LIL, by Ross Maller.- Commentary: Chris Heyde on Branching Processes and Population Genetics, by Eugene Seneta.- C. C. Heyde. On a property of the lognormal distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. , 25:392–393, 1963. Reprinted with permission of the Royal Statistical Society and John Wiley & Sons.- C. C. Heyde. Two probability theorems and their application to some first passage problems. J. Aust. Math. Soc. , 4:214–222, 1964. Reprinted with permission of the Australian Mathematical Society.- C. C. Heyde. Some renewal theorems with application to a first passage problem. Ann. Math. Statist. , 37:699–710, 1966. Reprinted with permission of the Institute of Mathematical Statistics.- C. C. Heyde. Some results on small-deviation probability convergence rates for sums of independent random variables. Canad. J. Math. , 18:656–665, 1966. Reprinted with the permission of the Canadian Mathematical Society.- C. C. Heyde. A contribution to the theory of large deviations for sums of independent random variables. Z. Wahrsch. Verw. Gebiete. , 7:303–308, 1967. Reprinted with permission of Springer Science+Business Media.- C. C. Heyde. On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Statist. , 38:1575–1578, 1967. Reprinted with permission of the Institute of Mathematical Statistics.- C. C. Heyde. On the influence of moments on the rate of convergence to the normal distribution. Z. Wahrsch. Verw. Gebiete. , 8:12–18, 1967. Reprinted with permission of Springer Science+Business Media.- C. C. Heyde. On large deviation probabilities in the case of attraction to a non-normal stable law. Sankhy Ser. A , 30:253–258, 1968. Reprinted with permission of the Indian Statistical Institute.- C. C. Heyde. On the converse to the iterated logarithm law. J. Appl.Probab. , 5:210–215, 1968. Reprinted with permission of the Applied Probability Trust.- C. C. Heyde. A note concerning behaviour of iterated logarithm type. Proc. Amer. Math. Soc. , 23:85–90, 1969. Reprinted with permission of the American Mathematical Society.- C. C. Heyde. On extended rate of convergence results for the invariance principle. Ann. Math. Statist. , 40:2178–2179, 1969. Reprinted with permission of the Institute of Mathematical Statistics.- C. C. Heyde. On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Probab. , 6:419–429, 1969. Reprinted with permission of the Applied Probability Trust.- C. C. Heyde. Some properties of metrics in a study on convergence to normality. Z. Wahrsch. Verw. Gebiete. , 11:181–192, 1969. Reprinted with permission of Springer Science+Business Media.- C. C. Heyde. Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist. , 41:739–742, 1970. Reprinted with permission of the Institute of Mathematical Statistics.- C. C. Heyde. On the implication of a certain rate of convergence to normality. Z. Wahrsch. Verw. Gebiete. , 16:151–156, 1970. Reprinted with permission of Springer Science+Business Media.- C. C. Heyde. A rate of convergence result for the super-critical Galton-Watson process. J. Appl. Probab. , 7:451–454, 1970. Reprinted with permission of the Applied Probability Trust.- C. C. Heyde and B. M. Brown. On the departure from normality of a certain class of martingales. Ann. Math. Statist. , 41:2161–2165, 1970. Reprinted with permission of the Institute of Mathem
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