Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis
The theory of one parameter semigroups of bounded linear operators on Banach spaces has deep and far reaching applications to partial differential equations and Markov processes. Here we present some known elementary applications of operator semigroups to approximation theory, a new proof of the central limit theorem, and we give E. Nelson's rigorous interpretation of Feynman integrals. Our main tools are (i) a special case of the Trotter-Neveu-Kato approximation theorem, of which we give a new elementary proof, and (ii) P. Chernoff's product formula. © 1976 Springer-Verlag New York Inc.
Goldstein, J. (1976). Semigroup-theoretic proofs of the central limit theorem and other theorems of analysis. Semigroup Forum, 12 (1), 189-206. https://doi.org/10.1007/BF02195925