Probability limit theorems and some questions in fluid mechanics 1)
A number of problems in fluid mechanics which have been dealt with by making use of a central limit theorem (and asymptotic normality) are mentioned. A discussion of central limit theorems for stationary processes and the need for some form of asymptotic independence is given.The concepts of uniform ergodicity and strong mixing are introduced.An example of asymptotic nonnormality is given when the form of asymptotic independence is not sufficiently strong.The derivation of a new result indicating that uniform ergodicity is strong mixing when there is trivial tail field is briefly sketched.
KeywordsCentral Limit Theorem Asymptotic Normality Stationary Sequence Folk Theorem Asymptotic Independence
Unable to display preview. Download preview PDF.
- 1.Batchelor, G. K.The Theory of Homogeneous Turbulence, Cambridge (1953).Google Scholar
- 2.Cocke, W. J.“Turbulent hydrodynamic line-stretching: the random-walk limit”, Physics of Fluids, to be published.Google Scholar
- 3.Cogburn, R. “Conditional probability operators”, Ann. Math. Statist. 33, 634–658 (1962).Google Scholar
- 5.Kolmogorov, A. N. “Mecanique de la turbulence” (Colloque Intern. du CNRS a Marseille), J. Fluid Mech. 13, (1962).Google Scholar
- 6.Loeve, M. Probability Theory, 3rd Edition, Van Nostrand.Google Scholar
- 7.Lumley, J. L. Stochastic Tools in Turbulence, Academic (1970).Google Scholar
- 8.Nagaev, S. V. “Some limit theorems for stationary Markov chains”, Theory Probability Appl. 2, 378–406 (1957).Google Scholar
- 9.Rosenblatt, M. “A central limit theorem and a strong mixing condition”, Proc. Nat. Acad. Sci. U.S.A. 42, 43–47 (1956).Google Scholar
- 10.Rosenblatt, M.“Independence and dependence”, Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability, 431–443 (1960).Google Scholar
- 11.Rosenblatt, M.Markov Processes:Structure and Asymptotic Behavior, Springer (1971).Google Scholar
- 13.Sun, T. C. “Some further results on central limit theorems for non-linear functions of a normal stationary process”, J. Math. Mech. 14, 71–85 (1965).Google Scholar
- 14.Yaglom, A. M. “The influence of fluctuations in energy dissipation on the shape of turbulence characteristics in the inertial interval”, Soviet PhysicsDoklady 11, 26–29 (1966).Google Scholar