Annex-1: On The “Duality” between Extremes and Sums

Sums

Maxima

\(\mathrm{ S_{k}= \sum _{1}^{k}X_{i} }\)

\(\mathrm{ M_{k}=\mathrm{ \begin{array}{c} \mathrm{ k} \\ \mathrm{ { max\\1} } \end{array} } \{ X_{i} \} }\)

\(\mathrm{ \varphi x \left( t \right) =M_{X}\left( e^{itX} \right) :ch.f.of~X }\)

\(\mathrm{ F_{X} \left( t \right) =Prob~ \{ X \leq x \} :d.f.~of~X }\)

\(\mathrm{ X_{i}~indep.:\varphi _{S_{k}} \left( t \right) = \prod_{1}^{k}\varphi _{i} \left( t \right) }\)

\(\mathrm{ X_{i}~indep.:F_{M_{k}} \left( x \right) = \prod_{1}^{n}F_{i} \left( x \right) }\)

\(\mathrm{ X_{i}~i.i.d.: \varphi _{S_{k}} \left( t \right) = \varphi^{k} \left( t \right) }\)

\(\mathrm{ X_{i}~i.i.d.:F_{M_{k}} \left( x \right) =F^{k} \left( x \right) }\)

\(\mathrm{ \left( X,Y \right) ~indep: \varphi aX+bY \left( t \right) = \varphi X \left( a~t \right) ~ \varphi Y \left( b~t \right) }\)

\(\mathrm{ \left( X,Y \right) ~indep:F_{max \left( X+a,Y+b \right) } }\)

\(\mathrm{ =F_{X} \left( x-a \right) F_{Y} \left( x-b \right) }\)

\(\mathrm{ \left( +~,~. \right) }\)

\(\mathrm{ \left( max~,~+ \right) }\)

\(\mathrm{ M \left( aX+b \right) =a~M \left( X \right) +b }\)

  \(\mathrm{ \cdots }\)

\(\mathrm{ V \left( aX+b \right) =a^{2}~V \left( X \right) }\)

  \(\mathrm{ \cdots }\)

If\(\mathrm{ \{ X_{i} \} }\) i.i.d. have \(\mathrm{ \mu ,~ \sigma ^{2} }\) then

\(\mathrm{ \mathrm{ \begin{array}{c} \mathrm{ \varphi \left( t \right) \\{ ( s_{k}-{k \mu })/{\sqrt[]{k}}~ \sigma }} \\ \end{array} } \rightarrow \left( e^{-t^{2}/2} \right) }\)

(Central Limit Theorem); in the general case “sometimes” the ch. f. of the normal law may be substituted by that of an indefinitely divisible law ; \(\mathrm{ \mu = \varphi_{X}^{’}{ \left( 0 \right) }/{i},~ \sigma ^{2}= \varphi _{X}^{’} \left( 0 \right) ^{2}- \varphi _{X}^{"} \left( 0 \right) }\), in the usual case.

If\(\mathrm{ \{ X_{i} \} }\)are i.i.d. there “sometimes” exist \(\mathrm{ \left( \lambda _{k},~ \delta _{k}>0 \right) }\) such that \(\mathrm{ Prob \{ \left( M_{k}- \lambda _{n} \right) / \delta _{n} \leq x \} =F^{k} \left( \lambda _{k}+ \delta _{k}~x \right) \rightarrow \tilde{L} \left( x \right) ,~\tilde{L} \left( x \right) }\)then being \(\mathrm{ \Psi _{ \alpha } \left( x \right) ~,~ \Lambda \left( x \right) ~or~ \Phi _{ \alpha } \left( x \right) }\); for \(\mathrm{ ~ \Lambda \left( x \right) ~ }\)we have \(\mathrm{ n \left( 1-F \left( \lambda _{k} \right) \right) \rightarrow 1,k \left( 1-F \left( \lambda _{k}+ \delta _{k} \right) \right) \rightarrow e^{-1} }\) or \(\mathrm{ \delta _{n} \sim {1}/{n}\,F’ \left( \lambda _{n} \right) }\); there are corresponding results for \(\mathrm{ \Phi _{ \alpha } }\) and \(\mathrm{ \Phi _{ \alpha } }\).

If \(\mathrm{ \left( X,Y \right) }\) has a binormal distribution standard normal margins and correlation coefficient then \(\mathrm{ Z=aX+bY }\) has a normal distribution \(\mathrm{ N \left( x/ \sigma \left( a,b \right) \right) }\)with \(\mathrm{ \sigma \left( a,b \right) =a^{2}+b^{2}+2~ \rho ~(a~b ) }\).

If \(\mathrm{ \left( X,Y \right) }\) has a bivariate distribution with reduced Gumbel margins then \(\mathrm{ Z=max \left( X-a~,~Y-b \right) }\) has a Gumbel distribution \(\mathrm{ \Lambda \left( z- \lambda \left( a~,~b \right) \right) }\) with \(\mathrm{ \lambda \left( a,b \right) =log~ \{ \left( e^{-a}+e^{-b} \right) k \left( b-a \right) \} }\)

If \(\mathrm{ \rho =0 }\)(independence), \(\mathrm{ \sigma \left( a,b \right) =1 }\) iff \(\mathrm{ a^{2}+b^{2}=1 }\); in the case \(\mathrm{ C \left( X~,~Z \right) =a }\).

If \(\mathrm{ k \left( w \right) =1 }\) (independence), \(\mathrm{ \lambda \left( a,b \right) =0 }\)  if  \(\mathrm{ e^{-a}+e^{-b}=1 }\); in that case \(\mathrm{ Prob~ \{ Z \leq X-a \} =Prob~ \{ X-b~ \leq X-~a \} =e^{-a} }\).