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Brownian Motion—Wiener Process

Published online by Cambridge University Press:  20 November 2018

Miklós Csörgő
Miklós Csörgő*
Affiliation:
Department of Mathematics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
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Bachelier (1900), Einstein (1905) and Smoluchowski (1915) provided a theory of the peculiar erratic motion of small particles suspended in a liquid, first described in 1826 by the English botanist Brown. In a series of papers beginning in 1920 Wiener undertook a mathematical analysis of Brownian motion.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 3 , 01 September 1979 , pp. 257 - 279
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bachelier, L., Théorie de la spéculation, Ann. Sci. Ecole Norm. Sup. 17 (1900), 21-86.Google Scholar
2. Book, S. A. and Shore, T. R., On large intervals in the Csörgő-Révész theorem on increments of a Wiener process, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 46 (1978), 1-11.Google Scholar
3. Chan, A. H. C., Csörgő, M. and Révész, P., Strassen type limit points for moving averages of a Wiener process, Can. J. Statist. 6 (1978), 57-75.Google Scholar
4. Chung, K. L., On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. Soc. 64 (1948), 205-233.Google Scholar
5. Chung, K. L., A Course in Probability Theory, Harcourt, Brace & World, New York, 1968.Google Scholar
6. Csâki, E. and Révész, P., How big must be the increments of a Wiener process? (1979), to appear.Google Scholar
7. Csörgő, M. and Révész, P., How big are the increments of a Wiener process? Ann. Probability, (1979a), to appear.Google Scholar
8. Csörgő, M. and Révész, P., How small are the increments of a Wiener process? Stochastic Processes and their Applications, 8 (1979b), 119-129.Google Scholar
9. Csörgő, M. and Révész, P., Strong Approximations in Probability and Statistics, (1979c), Book manuscript in progress.Google Scholar
10. Deo, C. M., A note on increments of a Wiener process, Technical Report, University of Ottawa, 1977.Google Scholar
11. Doob, J. L., Stochastic Processes, John Wiley, New York, 1953.Google Scholar
12. Einstein, A., On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat, Ann. Physik, 17 (1905), 549-560.Google Scholar
13. Erdôs, P. and Rényi, A., On a new law of large numbers, J. Analyse Math., 13 (1970), 103-111.Google Scholar
14. Feller, W., An Introduction to Probability Theory and its Applications I, John Wiley, New York, 1968.Google Scholar
15. Hirsh, W. M., A strong law for the maximum cumulative sum of independent random variables, Comm. Pure Appl. Math., 18 (1965), 109-127.Google Scholar
16. Itô, K., Stochastic integral, Proc. Imperial Acad., Tokyo, 20 (1944), 519-524.Google Scholar
17. Itô, K. and Nisio, M., On the oscillation functions of Gaussian processes, Math. Scand., 22 (1968), 209-223.Google Scholar
18. Kolmogorov, A. N., Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin, 1933.Google Scholar
19. Levy, P., Théorie de VAddition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.Google Scholar
20. Levy, P., Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1948.Google Scholar
21. McKean, H. P. Jr. Stochastic Integrals, Academic Press, New York, 1969.Google Scholar
22. Paley, R. E. A. C. and Wiener, N., Fourier Transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., Vol. 19, Amer. Math. Soc, Providence, R.I., 1934.Google Scholar
23. Paley, R. E. A. C., Wiener, N. and Zygmund, A., Note on random functions, Math. Z., 37 (1959), 647-668.Google Scholar
24. Smoluchowski, M., Über Brownsche Molekular-bewegung, Ann. Phys., 48 (1915), 110-120.Google Scholar
25. Strassen, V., An invariance principle for the law of interated logarithm, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 3, (1964), 211-226.Google Scholar
26. Wiener, N., The mean of a functional of arbitrary elements, Ann. of Math., 22 (1920), 66-72.Google Scholar
27. Wiener, N., Differential space, J. Math and Phys., 2 (1923), 132-174.Google Scholar
28. Wiener, N., Generalized harmonic analysis, Acta Math. 55 (1930), 117-258.Google Scholar
29. Wiener, N., I am a mathematician. The later life of a prodigy, Doubleday, New York, 1956.Google Scholar