On normal domination of (super)martingales
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | On normal domination of (super)martingales |
| 2. | Creator | Author's name, affiliation, country | Iosif Pinelis; Michigan Technological University |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | supermartingales; martingales; upper bounds; probability inequalities; generalized moments |
| 3. | Subject | Subject classification | Primary: 60E15, 60J65; secondary: 60E05, 60G15, 60G50, 60J30 |
| 4. | Description | Abstract | Let $(S_0,S_1,\dots)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $(H_{\le0},H_{\le1},\dots)$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that for every $i=1,2,\dots$ there exist $H_{\le(i-1)}$-measurable r.v.'s $C_{i-1}$ and $D_{i-1}$ and a positive real number $s_i$ such that $C_{i-1}\le X_i\le D_{i-1}$ and $D_{i-1}-C_{i-1}\le 2 s_i$ a.s. Then for all real $t$ and natural $n$ and all functions $f$ satisfying certain convexity conditions $ E f(S_n)\le E f(sZ)$, where $f_t(x):=\max(0,x-t)^5$, $s:=\sqrt{s_1^2+\dots+s_n^2}$, and $Z\sim N(0,1)$. In particular, this implies $ P(S_n\ge x)\le c_{5,0}P(sZ\ge x)\quad\forall x\in R$, where $c_{5,0}=5!(e/5)^5=5.699\dots.$ Results for $\max_{0\le k\le n}S_k$ in place of $S_n$ and for concentration of measure also follow. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | |
| 7. | Date | (YYYY-MM-DD) | 2006-11-21 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ejp.ejpecp.org/article/view/371 |
| 10. | Identifier | Digital Object Identifier | 10.1214/EJP.v11-371 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Journal of Probability; Vol 11 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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