Summary
The asymptotic behaviour of the stochastic process \(k \to \frac{1}{k}\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}k} {X_i }\) as k→∞-X ={X i : i=1,2,⋯ being a sequence of independent random variables having mean 0 and positive finite variance, satisfying both Lindeberg's condition and the strong law of large numbers — is studied by means of a distribution invariance principle. This invariance principle sharpens the classical one due to Donsker and Prokhorov describing the “weak” asymptotic behaviour of partial sums of independent random variables on a semi-infinite time interval. The topology of the path space being appropriately chosen it allows to compute the limit distributions of certain functionals associated to X, such as
Moreover, for uniformly bounded variables X i , a general estimate of the rapidity of convergence is derived and applied to various special cases
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Chung, K. L.: On the maximum partial sums of sequences of independent random variables. Trans. Amer. math. Soc. 64, 205–233 (1948).
Donsker, M. D.: An invariance principle for certain probability limit theorems. Mem. Amer. math. Soc. 6, 1–12 (1951).
Freedman, D. A.: Some invariance principles for functionals of a Markov chain. Ann. math. Statistics 38, 1–7 (1967).
Ito, K., and H. Mc Kean: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965.
Krickeberg, K.: Wahrscheinlichkeitsoperatoren von Verteilungen in VektorrÄumen. Trans. third Prague Conf. Information Theory, statist. Decision Functions, Random Processes 1962, 441–452 (1964).
Müller, D. W.: Non-standard proofs of invariance principles in probability theory. (Im Erscheinen begriffen.)
Prokhorov, Yu. V.: Convergence of random processes and limit theorems in probability theory. Theor. Probab. Appl. 1, 157–214 (1956).
Skorokhod, A. V.: A limit theorem for sums of independent random variables. Soviet Math. Dokl. 1, 810–811 (1960).
—: A limit theorem for homogeneous Markov chains. Theor. Probab. Appl. 8, 61–70 (1963).
—: Studies in the theory of random processes. Reading, Mass.: Addison-Wesley 1965.
Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. math. Statist. Probability, 315–343.
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Müller, D.W. Verteilungs-invarianzprinzipien für das starke gesetz der gro\en zahl. Z. Wahrscheinlichkeitstheorie verw Gebiete 10, 173–192 (1968). https://doi.org/10.1007/BF00531847
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DOI: https://doi.org/10.1007/BF00531847