Uniform estimate for distributions of the sum of i.i.d. random variables with fat tail (Q2902437)

This article focuses on the asymptotic estimate for the distribution of the partial sum of the sequence of i.i.d.r.v. Its special features include the fact that it is uniform and the type of large deviations. In addition these estimates have been recently used to calculate VaR, this popular measure of market risk. In a nutshell, this work by strenthening the assumptions specifies one of the estimates of Nagaeva. More specifically, the authors recall the following result of Nagaeva: NEWLINE\[NEWLINE\sup_{s\in [1,\infty)}\left|\frac{P(\sum_{k=1}^n X_k>sn^{1/2})} {\Phi_0(s)+\bar{F}(sn^{1/2})}-1\right|\to 0,\;\;\;n\to \infty,NEWLINE\]NEWLINE where i.i.d.r.v. \(X_n,\;n=1,2,\cdots,\) defined on a probability space \((\Omega,\mathcal{F},P),\) satisfy (\(\Phi_0(x)=P(N(0,1)>x)\)):NEWLINENEWLINE (A1) \(EX^2=1,\;EX=0;\) \(F(x)=P(X_1\leq x),\;\bar{F}(x)=1-F(x);\)NEWLINENEWLINE (A2) \(\bar{F}(x)\) is a regularly varying function of index \(-\alpha\) for some \(\alpha>2,\) as \(x\to \infty,\) i.e., if we let \(L(x)=x^{\alpha}\bar{F}(x),\;\;x\geq 1,\) then \(L(x)>0\) for any \(x\geq 1,\) and for any \(a>0\) NEWLINE\[NEWLINE\frac{L(ax)}{L(x)}\to 1,\;\;\;x\to \infty;NEWLINE\]NEWLINE (A3) \(|x|^{\alpha+2}F(x)\to 0,\;\;\;x\to -\infty.\)\NEWLINENEWLINEIn this work they prove three theorems, the first of which is the main, and the third offers a better approximation for \(P(\sum_{k=1}^n X_k>sn^{1/2})\) than \(\Phi_0(s)+\bar{F}(sn^{1/2}).\)