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2010 Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates
Adam Osekowski
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Electron. Commun. Probab. 15: 508-521 (2010). DOI: 10.1214/ECP.v15-1582

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.

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Adam Osekowski. "Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates." Electron. Commun. Probab. 15 508 - 521, 2010. https://doi.org/10.1214/ECP.v15-1582

Information

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

zbMATH: 1226.60060
MathSciNet: MR2733375
Digital Object Identifier: 10.1214/ECP.v15-1582

Subjects:
Primary: 60G42
Secondary: 60G44

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