An extremal property of independent random variables
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- by Steven Rosencrans and Stanley Sawyer PDF
- Proc. Amer. Math. Soc. 36 (1972), 552-556 Request permission
Abstract:
In a previous paper the first author proved $Ef(\smallint _0^t {e db) \leqq Ef(M{b_t})}$, where e is a Brownian functional $\leqq M$ 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: $Ef(\sum \nolimits _1^n {{d_k}{y_k}) \leqq Ef(M\sum \nolimits _1^n {{y_k})} }$. (The ${y_j}$’s are independent variables with mean zero, $j \to {d_1}{y_1} + \cdots + {d_j}{y_j}$ is a martingale, and $0 \leqq {d_j} \leqq M$.) We further show that these inequalities are false if t or n is a stopping time, or if ${d_j} \ngtr 0$.References
- D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
- Morris L. Eaton, A note on symmetric Bernoulli random variables, Ann. Math. Statist. 41 (1970), 1223–1226. MR 268930, DOI 10.1214/aoms/1177696897
- H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
- P. Warwick Millar, Martingales with independent increments, Ann. Math. Statist. 40 (1969), 1033–1041. MR 243605, DOI 10.1214/aoms/1177697607
- Steven Rosencrans, An extremal property of stochastic integrals, Proc. Amer. Math. Soc. 28 (1971), 223–228. MR 275535, DOI 10.1090/S0002-9939-1971-0275535-7
- Stanley Sawyer, Rates of convergence for some functionals in probability, Ann. Math. Statist. 43 (1972), 273–284. MR 301782, DOI 10.1214/aoms/1177692720
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 36 (1972), 552-556
- MSC: Primary 60G45; Secondary 60G50, 60J65
- DOI: https://doi.org/10.1090/S0002-9939-1972-0312566-3
- MathSciNet review: 0312566