Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above
| dc.creator | Pinelis, Iosif | |
| dc.date | 2005-12-14 | |
| dc.date | 2007-01-04 | |
| dc.date.accessioned | 2026-07-07T07:38:23Z | |
| dc.date.available | 2026-07-07T07:38:23Z | |
| dc.description | Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $σ$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf {Var}(X_i|H_{\le i-1})\le σ_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $σ_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where $Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf {Var}Z_i=σ^2:=\frac{1}{n}(σ_1^2+...+σ_n^2)$. Then, based on a comparison inequality between generalized moments of $S_n$ and $T_n$ for a rich class of generalized moment functions, the tail comparison inequality $$ \mathsf P(S_n\ge y) \le c \mathsf P^{\mathsf Lin,\mathsf L C}(T_n\ge y+\tfrach2)\quad\forall y\in \mathbb R$$ is obtained, where $c:=e^2/2=3.694...$, $h:=d+σ^2/d$, and the function $y\mapsto \mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto \mathsf {P}(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in \mathbb {Z}$). An explicit formula for $\mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y+\tfrac{h}{2})$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds. | |
| dc.description | Published at http://dx.doi.org/10.1214/074921706000000743 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0512301 | |
| dc.identifier | http://arxiv.org/abs/math/0512301 | |
| dc.identifier | IMS Lecture Notes Monograph Series 2006, Vol. 51, 33-52 | |
| dc.identifier | doi:10.1214/074921706000000743 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121066 | |
| dc.subject | Probability | |
| dc.subject | 60E15, 60G42, 60G48, 60G50 (Primary) | |
| dc.title | Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above | |
| dc.type | text |