Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above

Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-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 $\mathsf {Var}(X_i|H_{\le i-1})\le \sigma_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $\sigma_i>0$ 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 $\mathsf {Var}Z_i=\sigma ^2:=\frac{1}{n}(\sigma_1^2+...+\sigma_n^2)$. 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 $c:=e^2/2=3.694...$, $h:=d+\sigma ^2/d$, 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.Comment: 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

On the nonuniform Berry--Esseen bound

Due to the effort of a number of authors, the value c_u of the absolute constant factor in the uniform Berry--Esseen (BE) bound for sums of independent random variables has been gradually reduced to 0.4748 in the iid case and 0.5600 in the general case; both these values were recently obtained by Shevtsova. On the other hand, Esseen had shown that c_u cannot be less than 0.4097. Thus, the gap factor between the best known upper and lower bounds on (the least possible value of) c_u is now rather close to 1. The situation is quite different for the absolute constant factor c_{nu} in the corresponding nonuniform BE bound. Namely, the best correctly established upper bound on c_{nu} in the iid case is about 25 times the corresponding best known lower bound, and this gap factor is greater than 30 in the general case. In the present paper, improvements to the prevailing method (going back to S. Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, a new method is presented, of a rather purely Fourier kind, based on a family of smoothing inequalities, which work better in the tail zones. As an illustration, a quick proof of Nagaev's nonuniform BE bound is given. Some further refinements in the application of the method are shown as well.Comment: Version 2: Another, more flexible and general construction of the smoothing filter is added. Some portions of the material are rearranged. In particular, now constructions of the smoothing filter constitute a separate section. Version 3: a few typos are corrected. Version 4: the historical sketch is revised. Version 5: two references adde

Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(\eta)$ for a certain r.v. $\eta$. The r.v. $\eta$ and the class $\mathcal{F}$ depend on the choice of one of the three settings. In particular, $(m/s)\eta$ has the binomial distribution with parameters $n$ and $p:=m^2/(ns)$ in setting (ii) and the Poisson distribution with parameter $\lambda:=m^2/s$ in setting (iii). One can also let $\eta$ have the normal distribution with mean $m$ and variance $s$ in any of these three settings. In each of the settings, the class $\mathcal{F}$ contains, and is much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $P(S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $E\exp\{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.Comment: Version 2: fixed a typo (p. 17, line 2) and added a detail (p. 17, line 9). Version 3: Added another proof of Lemma 3.2, using the Redlog package of the computer algebra system Reduce (open-source and freely distributed