On normal domination of (super)martingales
By Iosif Pinelis. 2005
49 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/df459d88c5afffb1dc5e057744a9f5e687db01cf
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Exponential inequalities for martingales with applications (2013, background): “Moreover, we also show that (18) significantly improves some recent inequalities of Bentkus [3] and Pinelis [30, 31] by adding an exponential decay factor in the case ||∑ni=1C(2) i−1||∞ < ∑n i=1 ||C2 i−1||∞; see (42) and Example 1 for details.”
- On the Bennett–Hoeffding inequality (2009, background): “is cannot be done even for n= 1. imsart ver. 2005/05/19 file: arxiv.tex date: September 13, 2018 Iosif Pinelis/On the Bennett-Hoeffding bound 10 Remark 2.13. Quite similarly to how it was done e.g. in [44, 46], it is easy to extend the results of Theorem 2.1, Propositions 2.2, 2.8, and 2.9, and Corol”
- Probability and moment inequalities under dependence (2013, methodology): “Pinelis (2006) applied domination technique to obtain moment inequalities for supermartingales.”