The Annals of Probability

Convergence in distribution of nonmeasurable random elements

Patrizia Berti and Pietro Rigo

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A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

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Ann. Probab., Volume 32, Number 1A (2004), 365-379.

First available in Project Euclid: 4 March 2004

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Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures 60A05: Axioms; other general questions

Convergence in distribution empirical process exchangeability extension finitely additive probability measure measurability


Berti, Patrizia; Rigo, Pietro. Convergence in distribution of nonmeasurable random elements. Ann. Probab. 32 (2004), no. 1A, 365--379. doi:10.1214/aop/1078415839.

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  • Berti, P. and Rigo, P. (1994). Coherent inferences and improper priors. Ann. Statist. 22 1177--1194.
  • Berti, P. and Rigo, P. (1996). On the existence of inferences which are consistent with a given model. Ann. Statist. 24 1235--1249.
  • Berti, P. and Rigo, P. (1999). Sufficient conditions for the existence of disintegrations. J. Theoret. Probab. 12 75--86.
  • Berti, P., Pratelli, L. and Rigo, P. (2004). Limit theorems for a class of identically distributed random variables. Ann. Probab. To appear.
  • Bhaskara Rao, K. P. S. and Bhaskara Rao, M. (1983). Theory of Charges. Academic, New York.
  • Bumby, R. and Ellentuck, E. (1969). Finitely additive measures and the first digit problem. Fund. Math. 65 33--42.
  • Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3 89--99.
  • Dubins, L. E. and Prikry, K. (1995). On the existence of disintegrations. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1613 248--259. Springer, Berlin.
  • Dudley, R. (1966). Weak convergence of measures on non separable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 109--126.
  • Dudley, R. (1967). Measures on nonseparable metric spaces. Illinois J. Math. 11 449--453.
  • Dudley, R. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
  • Girotto, B. and Holzer, S. (1993). Weak convergence of masses on normal topological spaces. Sankhyā Ser. A 55 188--201.
  • Heath, D. and Sudderth, W. D. (1978). On finitely additive priors, coherence, and extended admissibility. Ann. Statist. 6 333--345.
  • Heath, D. and Sudderth, W. D. (1989). Coherent inference from improper priors and from finitely additive priors. Ann. Statist. 17 907--919.
  • Kallianpur, G. and Karandikar, R. L. (1983). A finitely additive white noise approach to non linear filtering. Appl. Math. Optim. 10 159--185.
  • Kallianpur, G. and Karandikar, R. L. (1988). White Noise Theory of Prediction, Filtering and Smoothing. Gordon and Breach, London.
  • Karandikar, R. L. (1982). A general principle for limit theorems in finitely additive probability. Trans. Amer. Math. Soc. 273 541--550.
  • Karandikar, R. L. (1988). A general principle for limit theorems in finitely additive probability: The dependent case. J. Multivariate Anal. 24 189--206.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Pyke, R. and Shorack, G. (1968). Weak convergence of a two sample empirical process and a new approach to Chernoff--Savage theorems. Ann. Math. Statist. 39 755--771.
  • Regazzini, E. (1987). De Finetti's coherence and statistical inference. Ann. Statist. 15 845--864.
  • van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes. Springer, New York.