Large Deviations For Sums Of Heavy-tailed Random Variables Of Several New Classes

Since heavy-tailed distribution class has important function toward application probability fields , people have researched it for many years,many scholars deeply studied the large deviations for heavy-tailed random variables. Kluppelberg(1997) got large deviation result as follows:Suppose F ∈ ERV(-α,-β),(1 < α < β < ∞), if (0-1),(0-2) are satisfied , Then for any fixed γ> 0i.e. P(S(t) - μ(t) > x) ~ X(t)F(x), as t →∞ holds uniformly for x ≥γλ(t).It is shown in [2] that , suppose F ∈ C ,which is a bigger class than ERV, assume (0-1) and (0-2) are satisfied , the same result is also right .In the present paper we defined two new classes which named G and E . G includes class C ,Pareto distribution and Lognormal distribution etc., E includes G and Weibull distribution etc.,and got the following results:1.Suppose F ∈ G , then for any fixed γ> 0 ,P(S_n - ES_n > x) ~ nF(x), as n →∞holds uniformly for 2.Suppose F ∈ G , if (0-1) ,(0-2) are satisfied ,then for any fixed γ> 0i.e. P(S(t) - μ(t) > x) ~ X(t)F(x), as t→∞ holds uniformly for 3.Suppose F G E.there exists 0, 0 < (3 < l,then for any fixed 7 > 0 ,P{Sn - ESn > x)lim supnF{x)-1= 0i.e.P{Sn-ESn > x) n-s-ooholds uniformly for x@ >-fn .4.Suppose F E E , if (0-l),(0-2)are satisfied,then there exists /?, 0 < /? < l,for any fixed 7 > 0,we havelim supt->ooP(S(t) - is(t) > x)\(t)F(x)1= 0i.e.P(S(t) - fi(t) > X) ~ \{t)F{x) (t -4 OO)holds uniformly for x@ > jX(t).