Extreme Values with Very Heavy Tails

Let X1,X2,… be a sequence of positive i.i.d.r.v.’s with So = 0, Sn = X1+X2+…+Xn (n=1,2,…) and let τt 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.