1 Introduction
2 Main results and discussions
3 Applications in insurance
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For two positive functions \(f(x)\) and \(g(x)\), we write$$\begin{aligned} &f(x)\sim g(x)\quad \mbox{if } \lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1, \\ &f(x)\lesssim g(x)\quad \mbox{if } \limsup_{x\rightarrow\infty}\frac{f(x)}{g(x)} \leq1. \end{aligned}$$
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For two positive bivariate functions \(f(\cdot,\cdot)\) and \(g(\cdot,\cdot)\), we say that \(f(x,t)\lesssim g(x,t)\), as \(t\rightarrow\infty\), holds uniformly in \(x\in\Delta _{t}\neq\emptyset\), if$$\limsup_{t\rightarrow\infty}\sup_{x\in\Delta_{t}}\frac {f(x,t)}{g(x,t)} \leq1. $$
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For a distribution function \(F(x)\) with finite mean \(\mu>0\), set \(\overline{F}(x)\equiv1-F(x)\) as the corresponding survival function of it.
3.1 Precise large deviations
3.2 Moderate deviations
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\(a(t)< C t\) for t large enough and a positive constant C;
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\(\lim_{t\rightarrow\infty}\frac{a(t)}{a([t])}=1\);
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\(\lim_{t\rightarrow\infty}\frac{n(\log n)^{\alpha }}{a(n)^{\alpha}}=0, 1<\alpha< \min\{2, J^{+}_{F}\}\),