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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the behavior of the constant in a decoupling inequality for martingales
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by Paweł Hitczenko PDF
Proc. Amer. Math. Soc. 121 (1994), 253-258 Request permission

Abstract:

Let $({f_n})$ and $({g_n})$ be two martingales with respect to the same filtration $({\mathcal {F}_n})$ such that their difference sequences $({d_n})$ and $({e_n})$ satisfy \[ P({d_n} \geq \lambda |{\mathcal {F}_{n - 1}}) = P({e_n} \geq \lambda |{\mathcal {F}_{n - 1}})\] for all real $\lambda$’s and $n \geq 1$. It is known that \[ {\left \| {{f^ \ast }} \right \|_p} \leq {K_p}{\left \| {{g^ \ast }} \right \|_p},\quad 1 \leq p < \infty ,\] for some constant ${K_p}$ depending only on p. We show that ${K_p} = O(p)$. This will be obtained via a new version of Rosenthal’s inequality which generalizes a result of Pinelis and which may be of independent interest.
References
  • D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
  • Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
  • PawełHitczenko, Comparison of moments for tangent sequences of random variables, Probab. Theory Related Fields 78 (1988), no. 2, 223–230. MR 945110, DOI 10.1007/BF00322019
  • PawełHitczenko, Best constant in the decoupling inequality for nonnegative random variables, Statist. Probab. Lett. 9 (1990), no. 4, 327–329. MR 1047832, DOI 10.1016/0167-7152(90)90141-S
  • PawełHitczenko, Best constants in martingale version of Rosenthal’s inequality, Ann. Probab. 18 (1990), no. 4, 1656–1668. MR 1071816
  • S. Kwapień and W. A. Woyczyński, Tangent sequences of random variables: basic inequalities and their applications, Almost everywhere convergence (Columbus, OH, 1988) Academic Press, Boston, MA, 1989, pp. 237–265. MR 1035249
  • Shlomo Levental, A uniform CLT for uniformly bounded families of martingale differences, J. Theoret. Probab. 2 (1989), no. 3, 271–287. MR 996990, DOI 10.1007/BF01054016
  • S. V. Nagaev and I. F. Pinelis, Some inequalities for the distributions of sums of independent random variables, Teor. Verojatnost. i Primenen. 22 (1977), no. 2, 254–263 (Russian, with English summary). MR 0443034
  • I. F. Pinelis, Estimates for moments of infinite-dimensional martingales, Mat. Zametki 27 (1980), no. 6, 953–958, 990 (Russian). MR 580071
  • V. V. Sazonov, On the estimation of the moments of sums of independent random variables, Teor. Verojatnost. i Primenen. 19 (1974), 383–386 (Russian, with English summary). MR 0348839
  • Joel Zinn, Comparison of martingale difference sequences, Probability in Banach spaces, V (Medford, Mass., 1984) Lecture Notes in Math., vol. 1153, Springer, Berlin, 1985, pp. 453–457. MR 821997, DOI 10.1007/BFb0074966
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 253-258
  • MSC: Primary 60G42
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1176481-2
  • MathSciNet review: 1176481