Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sharp square-function inequalities for conditionally symmetric martingales

Author: Gang Wang
Journal: Trans. Amer. Math. Soc. 328 (1991), 393-419
MSC: Primary 60G42
MathSciNet review: 1018577
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a conditionally symmetric martingale taking values in a Hilbert space $ \mathbb{H}$ and let $ S(f)$ be its square function. If $ {\nu _p}$ is the smallest positive zero of the confluent hypergeometric function and $ {\mu _p}$ is the largest positive zero of the parabolic cylinder function of parameter $ p$, then the following inequalities are sharp:

$\displaystyle \Vert f \Vert _{p} \leq \nu_{p}\Vert S(f)\Vert _{p}$   if$\displaystyle \;0 < p \leq 2,$

$\displaystyle \Vert f \Vert _{p} \leq \mu_{p} \Vert S(f)\Vert _{p}$   if$\displaystyle \;p \geq 3,$

$\displaystyle \nu_{p}\Vert S(f)\Vert _{p}\; \leq\; \Vert f\Vert _p$   if$\displaystyle \; p \geq 2.$

Moreover, the constants $ \nu_p$ and $ \mu_p$ for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.

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

  • [1] M. Abramowicz and I. A. Stegun, Handbook of mathematical functions, National Bureau of Standards, 1964.
  • [2] Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
  • [3] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 0208647
  • [4] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 0365692
  • [5] Donald L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158 (1988), 75–94. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976214
  • [6] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 0440695
  • [7] Burgess Davis, On the 𝐿^{𝑝} norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), no. 4, 697–704. MR 0418219
  • [8] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • [9] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
  • [10] J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonals, Ann. Soc. Polon. Math. 16 (1937), 84-96 (pp. 308-318 of Collected Papers).
  • [11] P. Warwick Millar, Martingale integrals, Trans. Amer. Math. Soc. 133 (1968), 145–166. MR 0226721, 10.1090/S0002-9947-1968-0226721-8
  • [12] A. A. Novikov, On stopping times for Wiener processes, Theory Probab. Appl. 16 (1971), 449-456.
  • [13] A. A. Novikov, The moment inequalities for stochastic integrals, Teor. Verojatnost. i Primenen. 16 (1971), 548–551 (Russian, with English summary). MR 0288844
  • [14] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
  • [15] A. O. Pittenger, Note on a square function inequality, Ann. Probab. 7 (1979), no. 5, 907–908. MR 542143
  • [16] L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 (1967), 1912–1914. MR 0217879
  • [17] G. Wang, Some sharp inequalities for conditionally symmetric martingales, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1989.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G42

Retrieve articles in all journals with MSC: 60G42

Additional Information

Keywords: Martingale, conditionally symmetric martingale, dyadic martingale, square-function inequality, confluent hypergeometric function, parabolic cylinder function, Brownian motion, Haar function
Article copyright: © Copyright 1991 American Mathematical Society