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Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates


 
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1. Title Title of document Sharp tail inequalities for nonnegative submartingales and their strong differential subordinates
 
2. Creator Author's name, affiliation, country Adam Osekowski; University of Warsaw
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Submartingale; Weak-type inequality; Strong differential subordination
 
3. Subject Subject classification 60G42; 60G44
 
4. Description 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 $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ 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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7. Date (YYYY-MM-DD) 2010-10-26
 
8. Type Status & genre Peer-reviewed Article
 
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9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1582
 
10. Identifier Digital Object Identifier 10.1214/ECP.v15-1582
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 15
 
12. Language English=en
 
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