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Sums of pairwise quasi-asymptotically independent random variables with consistent variation. (English) Zbl 1181.62011
Let $$X_1\dots X_n$$ be dependent real-valued random variables with distribution functions $$F_1, \dots F_n$$, respectively. They are pairwise quasi-asymptotically dependent and heavy-tailed, i.e., $Ee^{rX_i}=\int^{\infty}_{-\infty}e^{rX}dF_i(x)=\infty\quad \text{for\;any}\;r>0.$ The heavy-tailed distributions are subexponential distributions. Let us denote this class by $$\varphi$$. In the paper, the tail asymptotic behavior of the sum $$S_n=X_1+\dots +X_n$$ is considered. The main statements are as follows. Theorem 3.1 states that for $$F_i\in\varphi, 1\leq i\leq n$$, $P\{S_n>x\}\sim \sum_{n=1}^n \big(1-Fi(x)\big) \quad \text{as}\;x\rightarrow\infty.$ A similar statement holds for $P\bigg(\sum_{k=1}^nW_kX_k>x\bigg) \quad \text{if}\quad W_k,\;1\leq k\leq n,$ are independent of $$\{X_i,\;1\leq i\leq n\}$$ and integer-valued, with $$EW^p_k$$ for some $$p$$ and $$F_i\in\varphi$$.
If the random variables $$\{X_i, 1\leq i\leq n\}$$ are identically distributed with distribution $$F$$ with extended regular variation, conditions for the relation $P\bigg\{\sum^\infty_{k=1}W_kX_k>x\bigg\}\sim \sum^\infty_{n=1}P\{W_kX_k>x\}\quad \text{as}\;\;x\rightarrow\infty$ are obtained as well. Finally, if $$F\in\varphi$$ and $$\tau$$ is independent of the sequence $$\{X_1, X_2,\dots\}$$, integer-valued and such that $$0<E\tau^{p+1}<\infty$$ for some $$p$$, then $P\{S_\tau >x\}\sim E\tau\big( 1-F(x)\big), \quad \text{as}\quad x\rightarrow\infty.$ The Introduction and Chapter 2 contain references on the tail behavior of sums of dependent random variables as well as a discussion on heavy-tailed distributions. The list of references contains 33 positions.

##### MSC:
 62E20 Asymptotic distribution theory in statistics 60G50 Sums of independent random variables; random walks 62G32 Statistics of extreme values; tail inference 62H20 Measures of association (correlation, canonical correlation, etc.)
QRM
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