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A limit theorem for expectations conditional on a sum

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Abstract

Let ξ1, ξ2, ξ3,... be a sequence of independent random variables, such that μ j ≕E[ξ j ], 0<α⩽Var[ξ j ] andE[|ξ j −μ j |2+δ] for some δ, 0<δ⩽1, and everyj⩾1. IfU and ξ0 are two random variables such thatE[ξ 20 ]<∞ andE[|U|ξ 20 ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ j :j⩾1}, then under appropriate regularity conditions

$$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$

whereS n ≕ξ1+ξ2+⋯+ξ n ,μ j ≕E[ξ j ],s 2n ≕Var[S n ], andc n =O(s n ).

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References

  1. Bartlett, M. S. (1938). The characteristic function of a conditional statistic.J. London Math. Soc. 13, 62–67.

    Google Scholar 

  2. Billingsley, P. (1979).Probability and Measure, Wiley, New York.

    Google Scholar 

  3. Chung, K. L. (1974).A Course in Probability Theory, Second Edition, Academic Press, New York.

    Google Scholar 

  4. Cramér, H. (1970).Random Variables and Probability Distributions, Third Edition, Cambridge University Press, Cambridge.

    Google Scholar 

  5. Feller, W. (1971).An Introduction to Probability Theory and its Applications, Vol. 2, Second Edition, Wiley, New York.

    Google Scholar 

  6. Gnedenko, B. V., and Kolmogorov, A. N. (1954).Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  7. Khinchin, A. Ya. (1949).Mathematical Foundations of Statistical Mechanics, Dover, New York.

    Google Scholar 

  8. Kruglov, V. M. (1968). A local limit theorem for non-identically distributed random variables.Theor. Prob. Appl. 13, 332–334.

    Google Scholar 

  9. Liapunov, A. (1901). Nouvelle forme du théorème sur la limite des probabilités.Mém. Acad. Imp. Sci. St. Pétersbourg 12(5), 1–24.

    Google Scholar 

  10. Loève, M. (1977).Probability Theory 1, Fourth Edition, Springer-Verlag, New York.

    Google Scholar 

  11. Mazur, P., and van der Linden, J. (1963). Asymptotic form of the structure function for real systems.J. Math. Phys. 4, 271–277.

    Google Scholar 

  12. Petrov, V. V. (1975).Sums of Independent Random Variables. Springer-Verlag, New York.

    Google Scholar 

  13. Ray, D., and Sternberg, S. (1972). On the local limit theorem of classical statistical mechanics. (Unpublished manuscript.)

  14. Zabell, S. L. (1979). Continuous versions of regular conditional distributions.Ann. Prob. 7, 159–165.

    Google Scholar 

  15. Zabell, S. L. (1980). Rates of convergence for conditional expectations.Ann. Prob. 8, 928–941.

    Google Scholar 

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  1. Departments of Mathematics and Statistics, Northwestern University, 2006 Sheridan Road, 60208, Evanston, Illinois

    S. L. Zabell

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Zabell, S.L. A limit theorem for expectations conditional on a sum. J Theor Probab 6, 267–283 (1993). https://doi.org/10.1007/BF01047574

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