Electronic Communications in Probability

Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

Adam Osekowski

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Abstract

Let $f=(f_n)_{n\geq 0}$ be a nonnegative submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in[0,1]$ and $\beta\in \mathbb{R}$, the number $$L(x,y,t,\beta)=\inf\{||f||_1:\mathbb{P}(\sup_n g_n \ge\beta)\ge t\}$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 46, 508-521.

Dates
Accepted: 26 October 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243989

Digital Object Identifier
doi:10.1214/ECP.v15-1582

Mathematical Reviews number (MathSciNet)
MR2733375

Zentralblatt MATH identifier
1226.60060

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60G44: Martingales with continuous parameter

Keywords
Submartingale Weak-type inequality Strong differential subordination

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Osekowski, Adam. Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates. Electron. Commun. Probab. 15 (2010), paper no. 46, 508--521. doi:10.1214/ECP.v15-1582. https://projecteuclid.org/euclid.ecp/1465243989


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