Bounds on the expectation of functions of martingales and sums of positive RVs in terms of norms of sums of independent random variables

Author:
Victor H. de la Peña

Journal:
Proc. Amer. Math. Soc. **108** (1990), 233-239

MSC:
Primary 60E15; Secondary 60G42, 60G50

DOI:
https://doi.org/10.1090/S0002-9939-1990-0990432-9

MathSciNet review:
990432

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of random variables. Let be a sequence of independent random variables such that for each , has the same distribution as . If is a martingale and is a convex increasing function such that is concave on and then,

The same inequality holds if is a sequence of nonnegative random variables and is now any nondecreasing concave function on with . Interestingly, if is convex and grows at most polynomially fast, the above inequality reverses.

By comparing martingales to sums of independent random variables, this paper presents a one-sided approximation to the order of magnitude of expectations of functions of martingales. This approximation is best possible among all approximations depending only on the one-dimensional distribution of the martingale differences.

**[1]**D. R. Brillinger,*A note on the rate of convergence of a mean*, Biometrika**49**(3,4) (1962), 574. MR**0156372 (27:6295)****[2]**D. L. Burkholder, B. J. Davis, and R. F. Gundy,*Inequalities for convex functions of operators on martingales*, Proc. Sixth Berkeley Sympos. Math. Stat. Prob.**2**, 1972, pp. 223-240. MR**0400380 (53:4214)****[3]**S. W. Dharmadhikari and M. Sreehari,*On convergence in**-mean of normalized partial sums*, Ann. Probability**3**(6), (1975) 1023-1024. MR**0397869 (53:1725)****[4]**P. Hitczenko,*Comparison of mements for tangent sequences of random variables*, Probab. Theory Related Fields**78**(1988), 223-230. MR**945110 (90a:60089)****[5]**W. B. Johnson and G. Schechtman,*Sums at independent random variables in rearrangement invariant function spaces*, Ann. Probability**17**(2) (1989), 789-807. MR**985390 (90h:60045)****[6]**M. J. Klass,*Toward a universal law of the iterated logarithm, Part*I, Z. Wahrscheinlichkeithstheorie und verw. Gebiete**36**(1976), 165-178. MR**0415742 (54:3822)****[7]**-,*A method of approximating expectations of functions of sums of independent random variables*, Ann. Probability**9**(3) (1981), 413-428. MR**614627 (82f:60119)****[8]**B. von Bahr and C. G. Esseen,*Inequalities for the rth absolute moment of a sum of random variables*, , Ann. Math. Statist.,**36**(1965), 299-303. MR**0170407 (30:645)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60E15,
60G42,
60G50

Retrieve articles in all journals with MSC: 60E15, 60G42, 60G50

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-0990432-9

Keywords:
Martingales,
sums of independent random variables

Article copyright:
© Copyright 1990
American Mathematical Society