## Electronic Communications in Probability

### Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates

#### 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

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

Rights

#### 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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