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In 1945, very early in the history of the development of a rigorous analytical theory of probability, Feller (1945) wrote a paper called âThe fundamental limit theorems in probabilityâ in which he set out what he considered to be âthe two most important limit theorems in the modern theory of probability: the central limit theorem and the recently discovered ⦠âKolmogoroffâs cel ebrated law of the iterated logarithmâ â. A little later in the article he added to these, via a charming description, the âlittle brother (of the central limit theo rem), the weak law of large numbersâ, 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. Fellerâs introductory remarks include the visionary: âThe 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. |
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