Extreme Values with Very Heavy Tails

Abstract

Let X₁,X₂,… be a sequence of positive i.i.d.r.v.’s with S_o = 0, S_n = X₁+X₂+…+X_n (n=1,2,…) and let τ_t be the largest integer for which ${\operatorname{S} _{{{\tau _{t}}}}}\underline \leqslant \operatorname{t}$. Further let ${\operatorname{M} _{\operatorname{t} }}^{{\left( 1 \right)}}\underline \geqslant {\operatorname{M} _{t}}^{{\left( 2 \right)}}\underline \geqslant \cdots \underline \geqslant {\operatorname{M} _{t}}^{{\left( {{\tau _{t}} + 1} \right)}}$ be the torder statistics of the sequence ${X_1},{X_2},...,{X_\tau }_{_t},t - {S_\tau }_{_t}$ . The main result says that if IP(X₁< x)=exp(-(logx)^γ) (x≧1, 0< γ <1) then with proba-bility one for all t big enough ${\operatorname{t} ^{{ - 1}}}\left( {{\operatorname{M} _{t}}^{{\left( 1 \right)}} + {\operatorname{M} _{t}}^{{\left( 2 \right)}} + \cdots + {\operatorname{M} _{t}}^{{\left( \operatorname{r} \right)}}} \right)\underline \geqslant 1 - {\varepsilon _{t}}$if $\left( {\operatorname{r} - 2} \right)/\left( {\operatorname{r} - 1} \right)\underline \leqslant \gamma < \left( {\operatorname{r} - 1} \right)/\operatorname{r} ,{\varepsilon _{\operatorname{t} }}\underline \leqslant \exp \left( { - {{\left( {\log \operatorname{t} } \right)}^{\beta }}{{\left( {\log \log \operatorname{t} } \right)}^{{ - 3}}}} \right)$ and β=(r-1)/r-γ whenever (r-2)/(r-1) < γ <(r-1)/r.