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Inequalities for tails of adapted processes with an application to Wald's lemma

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Abstract

In this paper we introduce a new tail probability version of Wald's lemma for expectations of randomly stopped sums of independent random variables. We also make a connection between the works of Klass(18, 19) and Gundy(11) on Wald's lemma. In making the connection, we develop new Lenglart and Good Lambda inequalities comparing the tails of various types of adapted processes. As a consequence of our Good Lambda inequalities we include the following result. Let {d i }, {e i } be two sequences of variables adapted to the same increasing sequence of σ-fields ℱ n ↗ℱ, (e.g., ℱ n =σ({d i } n i=1 , {E i } n i=1 ), and letN⩽∞ be a stopping time adapted to {ℱ n }. Then for allp>0, there exists a constant 0<C p <∞ depending onp only, such that

$$\mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {d_i } } \right\| > \lambda } \right) \leqslant C_p \mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {e_i } } \right\| > \lambda } \right)$$

This result holds when the sequences are real, tangent, and either conditionally symmetric or nonnegative, or alternatively, if {d i } is a sequence of independent random variables and {e i } is an independent copy of {d i }, withN a stopping time adapted to the filtration generated by {d i } only. Other examples include Hilbert space valued differentially subordinate conditionally symmetric martingale differences. The result is true for more general operators applied to sequences as shown by an example comparing the square function of a conditionally symmetric sequence to the maximum of its absolute partial sums.

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References

  1. Azéma, J., Gundy, R. F., and Yor, M. (1979). Sur l'intégrabilité uniforme des martingales continues, Sém. de Probab. XIV, 1978–79.Lecture Notes in Math. 784, 53–61. Springer-Verlag, Berlin.

    Google Scholar 

  2. Blackwell, D. (1946). On an equation of Wald.Ann. Math. Statist. 17, 84–87.

    Google Scholar 

  3. Burkholder, D. L., and Gundy, R. F. (1970). Extrapolation and interpolation of quasilinear operators on martingales.Acta Math. 124, 249–304.

    Google Scholar 

  4. Burkholder, D. L. (1973). Distribution function inequalities for martingales.Ann. Prob. 1(1), 19–42.

    Google Scholar 

  5. Burkholder, D. L. (1979). Weak inequalities for exit times and analytic functions.Probability Theory Banach Center Publications Vol. 5. PWN-Polish Scientific Publishers, Warsaw.

    Google Scholar 

  6. Burkholder, D. L. (1988). Sharp inequalities for martingales and stochastic integrals.Astérisque 157–158, 75–94.

    Google Scholar 

  7. Burkholder, D. L. (1989). Differential subordination of harmonic functions and martingales. Proceedings of the seminar on harmonic analysis and partial differential equations (E1 Escorial, Spain).Lecture Notes in Math. 1384, 1–23.

    Google Scholar 

  8. Chow, Y. S., Robbins, H., and Siegmund, D. (1971).Great Expectations. The Theory of Optimal Stopping Houghton Mifflin, New York.

  9. de la Peña, V. H. (1990). Bounds on the expectation of functions of martingales and sums of positive rv's in terms of norms of sums of independent random variables.Proceedings Amer. Math. Soc. 108(1), 233–239.

    Google Scholar 

  10. de la Peña, V. H., and Govindarajulu, Z. (1990). A note on second moment of a randomly stopped sum of independent variables.Stat. Prob. Lett. 14, 275–281.

    Google Scholar 

  11. Gundy, R. F. (1981). On a theorem of F. and M. Riesz and an equation of A. Wald.Indiana Univ. Math. Journal 30(4), 589–605.

    Google Scholar 

  12. Hitczenko, P. (1988). Comparison of moments of tangent sequences of random variables.Prob. Th. Rel. Fields 78(2), 223–230.

    Google Scholar 

  13. Hitczenko, P. (1989). A remark on the paper “Martingale inequalities in rearrangement invariant function spaces,” by W. B. Johnson and C. Schechtman.Longhorn Notes. University of Texas at Austin Functional Analysis Seminar 1987-89.

  14. Hitczenko, P. (1990). Upper-bounds for theL p -norms of martingales.Prob. Th. Rel. Fields 86, 225–238.

    Google Scholar 

  15. Jacod, J. (1984). Une generalisation des semimartingales: Les processus admettant un processus a accroissements independants tangent.Lecture Notes in Math. 1247, 479–514.

    Google Scholar 

  16. Johnson, W. B., and Schechtman, G. (1988). Martingale inequalities in rearrangement invariant function spaces.Israel J. Math. 64, 267–275.

    Google Scholar 

  17. Kinderman, R. P. (1980). Asymptotic comparisons of functionals of Brownian Motion and Random Walk.Ann. Prob. 8(6), 1135–1147.

    Google Scholar 

  18. Klass, M. J. (1988). A best possible improvement of Wald's equation.Ann. Prob. 16(2), 840–853.

    Google Scholar 

  19. Klass, M. J. (1990). Uniform lower bounds for randomly stopped Banach space-valued random sums.Ann. Prob. 18(2), 790–809.

    Google Scholar 

  20. Krakowiak, W., and Szulga, J. (1988). On ap-stable multiple integral, II.Prob. Th. Rel. Fields 78(3), 449–453.

    Google Scholar 

  21. Kwapień, S., and Woyczyński, W. A. (1991). Semimartingale integrals via decoupling inequalities and tangent processes.Probability and Math. Statistics (to appear).

  22. Kwapień, S., and Woyczyński, W. A. (1989). Tangent sequences of random variables: Basic inequalities and their applications. InAlmost Everywhere Convergence, G. A. Edgar and L. Sucheston, eds., Academic Press, New York, pp. 237–265.

    Google Scholar 

  23. Lenglart, E. (1977). Relation de domination entre deux processus.Ann. Inst. Henri Poincaré 13, 171–179.

    Google Scholar 

  24. Lenglart, E., Lepingle, D., and Pratelli, M. (1979). Presentation Unifiée de certaines inégalités de la théorie des martingales. Sém. de Probab. XIV, 1978–79.Lecture Notes in Math. 784, 26–48. Springer-Verlag, Berlin.

    Google Scholar 

  25. McConnell, T., and Taqqu, M. (1984). Double integration with respect to symmetric stable processes. Tech. Report 618, Cornell University.

  26. McConnell, T., and Taqqu, M. (1986). Decoupling inequalities for multilinear forms in independent symmetric random variables.Ann. Prob. 14(3), 943–954.

    Google Scholar 

  27. McConnell, T., and Taqqu, M. (1987). Decoupling of Banach-valued multilinear forms in independent symmetric Banach-valued random variables.Prob. Th. Rel. Fields 75, 499–507.

    Google Scholar 

  28. Wald, A. (1945). Sequential tests of statistical hypotheses.Ann. Math. Statist. 16, 117–186.

    Google Scholar 

  29. Yor, M. (1979). Les inégalités des sous-martingales comme conséquences de la relation de domination.Stochastics 3, 1–15.

    Google Scholar 

  30. Zinn, J. (1985). Comparison of martingale difference sequences. Probability in Banach spaces V. Springer-Verlag, Berlin.Lecture Notes in Math. 1153, 453–457.

    Google Scholar 

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Authors and Affiliations

  1. Statistics Department, Columbia University, 10027, New York, New York

    Victor H. de la Peña

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  1. Victor H. de la Peña
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Supported in part by NSF Grants DMS 89-21369 and DMS 91-08006.

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de la Peña, V.H. Inequalities for tails of adapted processes with an application to Wald's lemma. J Theor Probab 6, 285–302 (1993). https://doi.org/10.1007/BF01047575

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  • DOI: https://doi.org/10.1007/BF01047575

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