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On the behavior of the constant in a decoupling inequality for martingales


Author: Paweł Hitczenko
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
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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

$\displaystyle P({d_n} \geq \lambda \vert{\mathcal{F}_{n - 1}}) = P({e_n} \geq \lambda \vert{\mathcal{F}_{n - 1}})$

for all real $ \lambda $'s and $ n \geq 1$. It is known that

$\displaystyle {\left\Vert {{f^ \ast }} \right\Vert _p} \leq {K_p}{\left\Vert {{g^ \ast }} \right\Vert _p},\quad 1 \leq p < \infty ,$

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 [Enhancements On Off] (What's this?)

  • [1] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42. MR 0365692 (51:1944)
  • [2] A. M. Garsia, Martingale inequalities, Seminar Notes on Recent Progress, Benjamin, Reading, MA, 1973. MR 0448538 (56:6844)
  • [3] P. Hitczenko, Comparison of moments for tangent sequences of random variables, Probab. Theory Related Fields 78 (1988), 223-230. MR 945110 (90a:60089)
  • [4] -, Best constant in the decoupling inequality for non-negative random variables, Statist. Probab. Letters 9 (1990), 327-329. MR 1047832 (91d:60104)
  • [5] -, Best constants in martingale version of Rosenthal's inequality, Ann. Probab. 18 (1990), 1656-1668. MR 1071816 (92a:60048)
  • [6] S. Kwapień and W. A. Woyczyński, Tangent sequences of random variables: basic inequalities and their applications, Proc. Conf. on Almost Everywhere Convergence in Probability and Ergodic Theory (Columbus, OH, 1988) (G. Edgar and L. Sucheston, eds.), Academic Press, San Diego, CA, 1989, pp. 237-265. MR 1035249 (91c:60020)
  • [7] S. Leventhal, A uniform CLT for uniformly bounded families of martingale differences, J. Theoret. Probab. 2 (1989), 271-287. MR 996990 (90f:60045)
  • [8] S. V. Nagaev and I. F. Pinelis, Some inequalities for the distributions of sums of independent random variables, Theory Probab. Appl. 22 (1977), 248-256. MR 0443034 (56:1407)
  • [9] I. F. Pinelis, Estimates of moments of infinite-dimensional martingales, Math. Notes 27 (1980), 459-462. MR 580071 (81k:60012)
  • [10] V. V. Sazonov, On the estimation of moments of sums of independent random variables, Theory Probab. Appl. 19 (1974), 371-374. MR 0348839 (50:1334)
  • [11] J. Zinn, Comparison of martingale differences,, Probability in Banach Spaces V (A. Beck, R. Dudley, M. Hahn, J. Kuelbs, and M. Marcus, eds.), Lecture Notes in Math., vol. 1153, Springer-Verlag, Berlin and New York, 1985, pp. 453-457. MR 821997 (87f:60070)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1176481-2
Keywords: Moment inequalities, martingale, tangent sequences
Article copyright: © Copyright 1994 American Mathematical Society

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