Hevy til nd stle distriutions J.K. Misiewicz Deprtment of Mthemtics nd Informtion Science Technicl University of Wrsw X or its distriution hs hevy til if E(X ) =. X or its distriution hs hevy til of order α, α (, ) if lim t t α P { X > t} = C <. E X p < i p < α. If you oserve lrge nd unexpected jumps etween mny rther smll oservtions you shll suspect hevy til distriution.
The pproximtion of the prmeter α we cn get using the following sttistic for t >> ( #{xk : x k > e t ), k =,..., n} F (t) = ln n αt + ln(c). The simplest exmples - Preto distriutions f(x) = αcα x α+ [c, )(x), or f(x) = αcα x α+ [c, )( x ). Fortuntely the densities re given explicitly, thus some of the technics of theoreticl sttistics cn e pplied. But, not ll of them!
3 Lw of Lrge Numers sttes tht if X, X,... re i.i.d hving til of order α > then X n = X + + X n n EX 5 5 Rysunek. W. Mtysik. X n for distriutions with nite expecttion In the cse α > this picture would look lmost the sme s for Gussin vriles.
If X, X,... re symmetric i.i.d hving til of order then X n = X + + X n n X, Ee itx = e A t 3 3 5 5 5 3 35 Rysunek. W. Mtysik. X n nd medin for symmetric distriution with the til order If α = this picture will not indicte ny convergence of the line X n.
5 If X, X,... re positive i.i.d hving til of order α < then X n = X + + X n n /α n ( α)/α.e. 6 5 3 5 5 Rysunek 3. W. Mtysik. X n for distriutions with nite expecttion If α <, nd only in this cse we cn hve positive stle vriles, the line X n would indicte convergence to innity. X + + X n n /α X α positive α-stle.
6 However the medin is converging to some constnt. One cn nd scle prmeter c > nd such tht the medin of the sequence Y n = (X n )/c converges to κ such tht ) Φ (κ / = 3 κ.98 6 5 3 5 5 Rysunek. W. Mtysik. Medin for distriutions with nite expecttion Prmeter cn e pproximted y inf {x k : k =,..., n}.
7 A rndom vrile X with the chrcteristic function ϕ(t) is stle if there re prmeters α (, ], σ >, β [, ] nd µ IR such tht the Fourier trnsform tkes the form ϕ(t) = { exp σ α t α( iβsgn(t) tn πα ) } + iµt if α ; { exp σ t ( + iβ π } sgn(t) ln t ) + iµt if α =. Four prmeters to e pproximted! We hve α. In the cse α > the prmeter µ is well pproximted y medin. If α then medin pproximtes some comintion of σ, β, µ.
8 Stle distriutions ply the sme role for hevy til models s Gussin distriution for distriutions with nite second moment, since Theorem. X is stle if it hs domin of ttrction, i.e. there exists sequence (Y n ) i.i.d rndom vriles, n >, n R such tht Y + + Y n n n d X. If X is α-stle then Y n hve hevy til distriution of order α. This mens tht whenever something cn e written s sum of very lrge numer of very smll pieces then it hs to hve stle distriution. Innite second moment, if α then lso rst moment does not exists. No explicit formul for the density function except α =,,. Sttistics for the prmeters α, β, σ, µ converge slowly. Literture: VERY rich list on the we-site of John Noln.
The dependence structure of stle rndom vectors (even in the symmetric cse) is much more complicted thn the dependence structure of Gussin rndom vector. 9
Su-gussin rndom vectors Ee i(x+y ) = exp { C ( + ) α/ }. exp( ( + ) / ) Contour plot 3.8.6.. 3 Rysunek 5. D. Kurowick. The chrcteristic function for su-gussin Cuchy rndom vector The sme level curves s for the Gussin rndom vector.
Stle rndom (X, Y) vectors with independent coordintes Ee i(x+y) = exp { C ( α + α )}. exp( ( + )) Contour plot 3.8.6.. 3 Rysunek 6. D. Kurowick. The chrcteristic function of Cuchy rndom vector with independent coordintes
α-stle β-su-stle rndom vector for < α < β <. (X, Y) = (Z, W) Θ /β, Ee i(x+y) = exp { C ( β + β) α/β }. exp( ( 3/ + 3/ ) /3 ) Contour plot 3.8.6.. 3 Rysunek 7. D. Kurowick. The chrcteristic function of 3/-su-stle Cuchy rndom vector
3 α-stle l β -symmetric rndom vector (X, Y) for α nd β >. Ee i(x+y) = exp { C ( β + β) α/β }. Possile only for two-dimensionl vectors! exp( ( + ) / ) Contour plot 3.8.6.. 3 Rysunek 8. D. Kurowick. The chrcteristic function of l -symmetric Cuchy rndom vector