An extremal property of independent random variables

Authors:
Steven Rosencrans and Stanley Sawyer

Journal:
Proc. Amer. Math. Soc. **36** (1972), 552-556

MSC:
Primary 60G45; Secondary 60G50, 60J65

MathSciNet review:
0312566

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a previous paper the first author proved , where *e* is a Brownian functional in absolute value and *f* is a convex function such that the right side is finite. We now prove a discrete analog of this inequality in which the integral is replaced by a martingale transform: . (The 's are independent variables with mean zero, is a martingale, and .) We further show that these inequalities are false if *t* or *n* is a stopping time, or if .

**[1]**D. L. Burkholder and R. F. Gundy,*Extrapolation and interpolation of quasi-linear operators on martingales*, Acta Math.**124**(1970), 249–304. MR**0440695****[2]**Morris L. Eaton,*A note on symmetric Bernoulli random variables*, Ann. Math. Statist.**41**(1970), 1223–1226. MR**0268930****[3]**H. P. McKean Jr.,*Stochastic integrals*, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR**0247684****[4]**P. Warwick Millar,*Martingales with independent increments*, Ann. Math. Statist.**40**(1969), 1033–1041. MR**0243605****[5]**Steven Rosencrans,*An extremal property of stochastic integrals*, Proc. Amer. Math. Soc.**28**(1971), 223–228. MR**0275535**, 10.1090/S0002-9939-1971-0275535-7**[6]**Stanley Sawyer,*Rates of convergence for some functionals in probability*, Ann. Math. Statist.**43**(1972), 273–284. MR**0301782**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60G45,
60G50,
60J65

Retrieve articles in all journals with MSC: 60G45, 60G50, 60J65

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1972-0312566-3

Keywords:
Martingale transform,
martingale,
stochastic integral,
convexity,
convex function,
inequalities

Article copyright:
© Copyright 1972
American Mathematical Society