Probabilities of large deviations for the \(\chi^2\) approximation
For a random variable \(X\) define cumulants \[ \Gamma_k(X) :=\frac{1}{i^k}\frac{d^k}{dt^k}\log{\mathbb E}e^{itX}|_{t=0},\quad k=1,2,\dots \] The authors define new coefficients (cumulants) \[ \hat{\Gamma}_k(X):=k!\sum_{n=1}^{k}(-1)^{k-n}2^{-n}C^{n-1}_{k-1} \frac{\Gamma_n(X)}{n!}, \quad k=1,2,\dots \] Let a random variable \(X=\chi^2_m\) have the \(\chi^2\)-distribution with degree of freedom \(m\geq 1\) and the characteristic function \(f(t)={\mathbb E}e^{it \chi^2_m}=(1-2it)^{-m/2}\). The main result of the paper is: Let a random variable \(X\) satisfy conditions \({\mathbb E}X={\mathbb E\chi^2_m}\), \[ |\hat{\Gamma}_k(X)-\hat{\Gamma}_k(\chi^2_m)|\leq \frac{k!}{\Delta^{k-1}}, \quad k=2,3,\dots \tag{1} \] for some \(\Delta>64\). Then for \(1<x<\Delta^{1/3}/\sqrt{2}\) we have large deviations inequality \[ {\mathbb P}(X\geq x)={\mathbb P}(\chi^2_m\geq x)\biggl(1+\theta\frac{x^3}{\Delta}\biggr), \] where \(|\theta|\leq 6.8 2^{m/2-1} \Gamma(m/2)\). In a future paper the authors plan to obtain a similar inequality under the weaker condition \[ |\hat{\Gamma}_k(X)-\hat{\Gamma}_k(\chi^2_m)|\leq \frac{(k!)^{1+\gamma}}{\Delta^{k-1}}, \quad k=2,3,\dots , \] where \(\gamma\geq 0\).
scientific article; zbMATH DE number 1296178
scientific article