J. Math. Sci. Uiv. Tokyo 8 (2, 397 427. Uiform Estimates for Distributios of the Sum of i.i.d. om Variables with Fat Tail i the Threshold Case By Keji Nakahara Abstract. We show uiform estimates for distributios of the sum of i.i.d. rom variables i the threshold case. ozovskii showed several uiform estimates but the speed of covergece was ot kow. Our mai uiform estimate implies a speed of covergece. We also compare our estimates with Nagaev s estimate which is valid i the o-threshold case, moreover, give a ecessary sufficiet coditio for Nagaev s estimate to hold i the threshold case.. Itroductio Let (Ω, F,Pbeaprobability space X, =, 2,..., be idepedet idetically distributed rom variables whose probability laws are µ. Let F : [, ] F : [, ] be give by F (x =µ((,x] F (x = µ((x,, x. Weassume the followig. (A F (x isaregularly varyig fuctio of idex α for some α 2, as x,i.e., if we let L(x =x α F (x,x, the L(x > for ay x, for ay a> L(ax L(x,x. (A2 x α+δ µ(dx < for some δ (,, x2 µ(dx = xµ(dx =. S.V. Nagaev [5] proved the followig theorem. 2 Mathematics Subject Classificatio. 6F5, 62E2. 397
398 Keji Nakahara Theorem (Nagaev. have ( sup s [, Assume (A for α>2 (A2. The we P ( k= X k > /2 s Φ (s+ F ( /2, s. Here Φ : is give by Φ (x = 2π x exp( y2 dy, x. 2 I this paper, we assume (A for α =2(threshold case, (A2 the followig. (A3 The probability law µ is absolutely cotiuous has a desity fuctio ρ : [, which is right cotiuous has a fiite total variatio. We show two uiform estimates. Our mai estimate gives the speed of covergece. The other oe is similar to (. Let us defie Φ k :,k =, 2, 3by Φ (x = exp( x2 2π 2 = d dx Φ (x, k dk Φ k (x =( dx k Φ (x, k =2, 3. Let v = /2 x2 µ(dx for. Our mai result is the followig. Theorem 2. Assume (A for α =2, (A2 (A3. The for ay δ (,, there is a costat C> such that (2 Here sup s [, H(, s = Φ (s+ ( P ( k= X k > /2 s H(, v /2 CL( /2 δ. s s v /2 /2 Φ (s F ((s xv /2 /2 Φ (xdx xµ(dx+v Φ 2 (s /2 x 2 µ(dx 2.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 399 We also show a similar uiform estimate to ( a ecessary sufficiet coditio for ( to hold uder the three assumptios. Theorem 3. Assume (A for α =2, (A2 (A3. The we have (3 sup s [, P ( k= X k > /2 s Φ (v /2 s+ F ( /2 s,. ozovskii [6] showed differet types of uiform estimate. (see Theorems, 2 3b i [6]. The estimates i Theorems 2 i [6] were proved uder more geeral assumptios but they were complex the speed of covergece was ot proved. The estimate i Theorem 3b i [6] is strogly related to (3 but does ot ecessarily imply our result. The proof of uiform estimates i [6] is differet from ours. We also prove the followig. Theorem 4. Assume (A for α = 2, (A2 (A3. If lim sup ( v log L( /2 =, the we have (4 sup s [, Φ (v /2 s+ F ( /2 s Φ (s+ F ( /2, s. If lim sup ( v log L( /2 >, the (4 does ot hold. Combiig Theorems 2 3 gives a ecessary sufficiet coditio for ( to hold, i.e. if we assume (A for α =2,(A2, (A3 lim sup ( v log L( /2 =,the (3 holds, amely P ( X k >s Φ ( /2 s+ F (s, for s> /2. k= The coditio lim sup ( v log L( /2 =correspods to (56 i [6]. Hece the estimate with B = /2 i Theorem 3b i [6] is ot valid uder our assumptios.
4 Keji Nakahara We also prove the followig to obtai Theorem 2. Theorem 5. Assume (A for α =2, (A2 (A3. The for ay δ (,, there is a costat C> such that P ( k= X k >s /2 H(, v /2 s CL( /2 2 δ, s. Throughout this paper we assume (A for α =2,(A2 (A3. The we see that L(t, t v L( /2, (see (5 (6. 2. Prelimiary Facts We summarize several kow facts (c.f. Fushiya-Kusuoka[2]. Propositio. We have L(ax sup /2 a 2 L(x, x, L(ax if /2 a 2 L(x, x. Propositio 2. For ay ε (,, there is a M(ε such that M(ε y ε L(yx L(x M(εyε x, y. Propositio 3. t β+ L(t (2 For ay β>, t β+ L(t ( For ay β<, t t x β L(xdx β +, x β L(xdx β +, t. t.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 4 (3 Let f :[, (, be give by f(t = t x L(xdx t. The f is slowly varyig. Moreover if lim t f(t <, wehave L(t t x L(xdx, t. Propositio 4. There is a costat C > such that Φ k (x C ( + x k Φ (x, x,k =, 2 C Φ (x xφ (x C Φ (x, x /2. Propositio 5. ( For ay m, let r e,m : C be give by The we have r e,m (t =exp(it ( + m (it k, t. k! k= r e,m (t mi( t m+, 2(m + t m, t. (m +! (2 For ay m, let r l,m : {z C; z /2} C be give by m ( k r l,m (z =log( + z z k, k k= z C, z /2. The we have r l,m (z 2 z m+, z C, z /2.
42 Keji Nakahara Let µ(t, ν(t, t>, be probability measures o (, B( give by µ(t(a = ( F (t µ(a (,t], ν(t(a = F (t µ(a (t, ], for ay A B(. Let ϕ( ; µ(t (resp. ϕ( ; ν(t, t>, be the characteristic fuctio of the probability measure µ(t (resp. ν(t,i.e., ϕ(ξ; µ(t = exp(ixξµ(t(dx, ξ. Propositio 6. There is a costat c > such that for ay t 2, ξ positive itegers, m with m, ϕ( /2 ξ; µ(t ( + c m ξ 2 m/4. Propositio 7. Let ν be aprobability measure o (, B( such that x2 ν(dx <. Also, assume that there is a costat C>such that the characteristic fuctio ϕ( ; ν : C satisfies The for ay x v> ϕ(ξ; ν C( + ξ 2, ξ. ν((x, = Φ (v /2 x+ 2π e ixξ iξ (ϕ(ξ,ν exp( vξ2 2 dξ. 3. Estimate for Momets Characteristic Fuctios Let η k (t = t η 3 (t = x k µ(dx, t >, k=, 2, t x 3 µ(dx, t >.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 43 The we see that η (t = t xµ(dx = t F (xdx + t F (t, t >, η 2 (t = t x 2 µ(dx =2 t x F (xdx + t 2 F (t, t >, t η 3 (t = F ( t 3 F (t+3 x 2 F (xdx t >. I particular, we see that (5 L(t η 2 (t, t, (6 η 2 (t L(t, t. For ay δ>, let t = /2 L( /2 δ. Note that /2 t,. Propositio 8. For ay ε>, there is a costat C> such that (7 (8 (9 ( ( L(t L( /2 CL(/2 εδ F (t CL( /2 2δ εδ η 2 ( /2 η 2 (t CL( /2 2εδ /2 η (t CL( /2 2δ /2 η 3 (t CL( /2 for ay. Proof. From Propositio 2, there is a M(ε > such that L(t L( /2 = L(t L(t L( /2 δ M(εL(/2 εδ.
44 Keji Nakahara Hece we have (7. Similarly, we see that F (t =L( /2 2δ L(t =L( /2 2δ L(t L( /2 η 2 ( /2 η 2 (t = L(t L( /2 +2 /2 t = L(t L( /2 +2L(t L(z z dz L(t δ L(t L( /2 +2L(t M(ε L(t y L(t y dy L(t δ y +ε dy L(t L( /2 +2 M(ε L(t (L( /2 εδ. ε Therefore by (7, we have (8 (9. Let ε (t = t x 2 L(xdx L(t t ε 3 (t = t L(xdx. tl(t The from Propositio 3 ( (2 we have ε (t ε 3 (t as t. Hece we see that /2 η (t = /2 (t F (t + t F (xdx = L( /2 δ L(t (2 + ε (t = (2+ε (t L( /2 δ L(t L( /2 /2 η 3 (t = /2 F (+(2+ε3 (t L( /2 δ L(t = /2 F (+(2+ε3 (t L( /2 +δ L(t L( /2. From (7, we have ( (.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 45 4. Asymptotic Expasio of Characteristic Fuctios emid that v = /2 x2 µ(dx t = /2 L( /2 δ.ithis sectio, we prove the followig lemma. Lemma. Let, (ξ = exp( v 2 ξ2 ( F (t ϕ( /2 ξ; µ(t ( + (( F (t ϕ( /2 ξ; µ(t + v 2 ξ2,, (ξ = exp( v 2 ξ2 ( F (t ϕ( /2 ξ; µ(t,,2 (ξ = exp( v 2 ξ2 ( F (t ϕ( /2 ξ; µ(t. The there is a costat C> such that (2, (ξ CL( /2 2 5δ ξ (3, (ξ +,2 (ξ CL( /2 2δ ξ for ay 8 ξ with ξ L( /2 δ. As a corollary to Lemma, we have the followig. Corollary. Let (, s = ( F (t µ(t ((s /2, Φ (v /2 s e isξ ((( 2π iξ F (t ϕ( /2 ξ; µ(t + v ξ 2 2 e v ξ 2 /2 dξ,k (, s = ( F (t k µ(t ( k ((s /2, Φ (v /2 s, k =,.
46 Keji Nakahara The there is a costat C> such that for ay, wehave for s (4 (, s CL( /2 2 6δ (5, (, s +, (, s CL( /2 4δ. Proof. From Propositio 8, we see that (, s = 2π iξ e isξ (( F (t ϕ( /2 ξ; µ(t e v ξ 2 ( ( ( F (t ϕ( /2 ξ; µ(t + v ξ 2 2 = e isξ 2π iξ,(ξe v ξ 2 /2 dξ. By Lemma, there is a costat C > such that, (ξ dξ C L( /2 2 6δ. ξ L( /2 δ ξ It is easy to see from Propositio 5 ( that 2 e v ξ 2 2 dξ ( F (t ϕ( /2 ξ; µ(t /2 η (t ξ + ξ 2 2, ξ. From the above iequality Propositio 7, we see that for ay m 2/δ, there is a costat C > such that for ay 2m ξ with ξ L( /2 δ, ϕ( /2 ξ; µ(t + (( F (t ϕ( /2 ξ; µ(t ++ v ξ 2 e v ξ 2 2 C ξ m. 2 Hece we have ξ ( F (t ϕ( /2 ξ; µ(t e v ξ 2 2 ξ >L( /2 δ
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 47 ((( F (t ϕ( /2 ξ; µ(t + v ξ 2 2 e v ξ 2 2 dξ 2C ξ m dξ = 2C L( /2 δ m L(/2 mδ 2C m L(/2 2. Therefore we have (4. We see also that,k (, s = e isξ 2π iξ = 2π ( ( F (t k ϕ( /2 ξ; µ(t k e v ξ 2 2 e isξ iξ,+k(ξe v ξ 2 /2 dξ. Similarly to the first equatio, we have (5. dξ We make some preparatios to prove Lemma. Let (, ξ =( F (t ϕ( /2 ξ,µ(t ( v ξ 2 2. First we prove the followig. Propositio 9. There is a costat C> such that for ay 8, ξ with ξ L( /2 δ, I particular, (, ξ CL( /2 2δ ξ ( F (t ϕ( /2 ξ; µ(t CL( /2 δ ξ. (6 sup{ (, ξ ; ξ L( /2 δ },.
48 Keji Nakahara Proof. We ca easily see that ϕ(ξ; µ(t = exp(ixξµ(t(dx = +η (t(iξ+η 2 (t (iξ2 + r e,2 (ξxµ(dx 2 t + r e,2 (ξxµ(dx+ F (t t F r e, (ξxµ(dx. (t Hece we have that ( F (t (, ξ = /2 η (t (iξ+(η 2 (t η 2 ( /2 (iξ2 2 + r e,2 ( /2 ξxµ(dx The we see that + t + F (t F (t r e,2 ( /2 ξxµ(dx t r e, ( /2 ξxµ(dx. (, ξ /2 η (t ξ +(η 2 ( /2 η 2 (t ξ 2 2 + δ /2 x 2+δ µ(dx ξ 2+δ + 6 /2 η 3 (t ξ 3 + /2 F (t x µ(dx ξ, ξ,t 2, where δ is i (A2. Hece from Propositio 5, we see that there is a costat C>such that (, ξ ( C L( /2 2δ ξ + L( /2 δ ξ 2 + δ/2 ξ 2+δ + L( /2 ξ 3. Therefore we have the first iequality.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 49 Sice ( ( F (t ϕ( /2 ξ; µ(t = (, ξ η 2 ( /2 ξ 2 /2, we have the secod iequality. For k =,, let,k (, ξ = ( k log ( ( F (t ϕ( /2 ξ; µ(t (( F (t ϕ( /2 ξ; µ(t. Propositio. There is a costat C>such that for ay ξ with ξ L( /2 δ k =,,,k (, ξ C L( /2 3δ ξ. I particular, for k =, we have (7 sup{,k (, ξ ; ξ L( /2 δ },. Proof. First, for ay ξ with ξ L( /2 δ,wehave log ( ( F (t ϕ(ξ,µ(t = ( F (t ϕ(ξ,µ(t +r l, (( F (t ϕ(ξ,µ(t. Hece we have,k (, ξ = k log ( ( F (t ϕ( /2 ξ; µ(t + r l, (( F (t ϕ( /2 ξ; µ(t. From Propositio 9, we see that there is a costat C>such that,k (, ξ ( F (t ϕ( /2 ξ; µ(t +2 ( F (t ϕ( /2 ξ; µ(t 2 C L( /2 3δ ξ, ξ L( /2 δ. Let us prove Lemma. Note that for k =, wehave log(e v ξ 2 /2 ( F (t k ϕ( /2 ξ; µ(t k = (, ξ+,k (, ξ.
4 Keji Nakahara We see that e v ξ 2 /2 ( F (t k ϕ( /2 ξ; µ(t k = exp( (, ξ+,k (, ξ. Hece we see that, (ξ = e v ξ 2 /2 ( F (t ϕ( /2 ξ; µ(t ( + (, ξ = exp( (, ξ ( + (, ξ + exp( (, ξ(exp(, (, ξ. By (6, we see that there is a costat C> such that, (ξ C ( (, ξ 2 +, (, ξ. Therefore we have (4 from Propositios 9. The proof of (5 is similar to (4. 5. Proof of Theorem 5 Note that P ( X l >s /2 = I k (, s, l= k= where I k (, s =P ( The we have I k (, s = = X l >s /2, l= ( P ( k {Xl >t } = k, k =,,...,. l= X l >s /2,X i >t,i=,...,k, l= X j t,j = k +,..., ( F (t k ( k F (t k µ(t ( k ν(t k (( /2 s,, for k =,,...,.Weestimate I (, s, I 2 (, s k=2 I k(, s oe by oe. This approach was origially used i A.V. Nagaev s papers ([3], [4].
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 4 Let F, (x =P (X > /2 x, X t =( F (t µ(t (( /2 x, F, (x =P (X > /2 x, X >t. Note that F, (x + F, (x = F ( /2 x. Propositio. There is a costat C> such that I (, s ( Φ (v /2 s 2 Φ 2(v /2 s F, (s v /2 xφ (xdx CL( /2 2 5δ,,s. Proof. First, we see that F, (s v /2 xφ (xdx Φ (v /2 s ( = e ixξ (( s 2π F (t ϕ( /2 ξ; µ(t e v 2 ξ 2 dξ = e isξ 2π iξ (( F (t ϕ( /2 ξ; µ(t e v 2 ξ 2 dξ. Hece we have ( F, (s v /2 xφ (xdx Φ (v /2 s + 2 Φ 2(v /2 s e isξ ((( F (t ϕ( /2 ξ; µ(t + v ξ 2 = 2π iξ By Corollary, we have our assertio. 2 dx e v ξ 2 /2 dξ. Propositio 2. There is a costat C> such that I (, s F, (s v /2 xφ (xdx CL( /2 2 6δ,,s. Proof. We see that I (, s = F (t ( F (t ν(t µ(t ( (( /2 s, = F (t ( F (t µ(t ( (( /2 (s /2 x, ν(t (dx
42 Keji Nakahara F, (s v /2 xφ (xdx = F (t ν(t (( /2 (s v /2 x, Φ (xdx = F (t ν(t (( /2 s x, Φ ( /2 v /2 x /2 v /2 dx = F (t Φ (v /2 s /2 v /2 xν(t (dx. Hece we have I (, s F, (s v /2 xφ (xdx = F ( (t ( F (t µ(t ( (( /2 s x, Φ (v /2 (s /2 x ν(t (dx. Therefore, by Corollary, we have our assertio. Let us prove Theorem 5. From Propositios 2, we see that there is a costat C> such that I (, s+i (, s ( Φ (v /2 s 2 Φ 2(v /2 s F ( /2 (s v /2 xφ (xdx CL( /2 2 6δ. Note that = F ( /2 (s v /2 xφ (xdx Φ (v /2 s /2 v s + v /2 s F ( /2 (s v /2 xφ (xdx ( F ( /2 (s v /2 x /2 {v x>s} Φ (xdx
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 43 = /2 v s v /2 s F ( /2 v /2 (v /2 s xφ (xdx F ( /2 (s v /2 xφ (xdx v /2 s = /2 F (( /2 (s v xφ (xdx F (yφ (v /2 s /2 v /2 yv /2 Let (z,y =Φ (z y Φ (z Φ 2 (zy, for z>,y, the we see that there is a costat C > such that Hece we have v /2 s 2 k= = /2 (s, y C y +δ. F ( /2 (s v xφ (xdx v k/2 k/2 Φ k (v /2 s (v /2 C δ /2 v (+δ /2 s, /2 v /2 yf (ydy y +δ F (ydy C δ/2, where C = C v (+δ /2 y+δ F (ydy <. Sice F (ydy = yµ(dy = dy. y k F (ydy yµ(dy yf(ydy = y 2 µ(dy = v 2 2 /2 y 2 µ(dy, 2
44 Keji Nakahara we see that = v Therefore we have We also see that 2 Φ 2(v /2 s+v Φ 2 (v /2 s Φ 2 (v /2 s /2 y 2 µ(dy. 2 y 2 F (ydy ( Φ (v /2 s+ 2 Φ 2(v /2 s + F ( /2 (s v /2 xφ (xdx H(, v /2 s C δ/2. I k (, s k=2 k=2 ( ( 2 F (t k ( k(k k 2 F (t k ( F (t 2 L( /2 2 5δ. 2 This completes the proof of Theorem 5. 6. Some Estimatios Let ˆF (s = s F ((s xv /2 A(, s = ˆF (s v /2 /2 Φ (s v /2 2 Φ 2(s = ˆF (s v /2 v Φ 2 (s /2 Φ (xdx, x 2 µ(dx, /2 Φ (s ( /2 xµ(dx F (xdx x F (xdx L(/2 2.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 45 The we have Let H(, s = Φ (s+a(, s. H (, s =Φ (s+ F (v /2 /2 s. I this sectio we prove the followig lemma. Lemma 2. sup s [, H(, s H (, s,. Let u = v /2 /2, α = L(u /3 β = L(u /2. Propositio 3. For ay ε>, there is a costat C> such that F (u s CL(u s 2+ε, s [,. I particular, for s>β we have F (u s Cs4+ε. Proof. From Propositio 8 we see that for ay ε> there is a costat C>such that = v s 2 L(u F (u s L(u L(u s CL(u s 2+ε. Sice L(u = β 2 s 2 for s>β,wehave the secod iequality. Let ˆF (s = 4 k= I k(, s, where I (, s = I 2 (, s = s s α 7/8s I 3 (, s = s F ((s xu Φ (xdx, s α 7/8s F ((s xu Φ (xdx, F ((s xu Φ (xdx
46 Keji Nakahara s I 4 (, s = F ((s xu Φ (xdx. Let (, s, y = Φ (s u y (Φ (s+u yφ 2 (s, for, s,y [,. Propositio 4. sup H (, s I (, s s [, 2 k= v k/2 (k 2/2 Φ k (s α u y k F (ydy, Proof. We see that 2 I (, s v k/2 (k 2/2 Φ k (s = u = u k= α u α u Note that for ay y [,α u ], Hece for all s [, F (y ( Φ (s u F (y(, s, ydy. α u y k F (ydy. y Φ (s u yφ 2 (s dy (, s, y u 2 y 2 sup Φ 3 (z z [s α,s] C y 2 ( + s 2 Φ (s α C 2 y 2 ( + s 3 Φ (s exp(α s. I (, s 2 k= α v k/2 (k 2/2 u Φ k (s y k F (ydy 8C sup{z 2 F (z; z }α s 3 Φ (s exp(α s.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 47 Sice α β 3 = L(u /2,,wehave sup Φ (s I (, s s β 2 k= α u v k/2 (k 2/2 Φ k (s y k F (ydy,. From Propositio 3, we see that for ay ε> there is a costat C(ε > such that Hece we see that for s>β, ( F (u s C(εs 4+ε. ( F (u s I (, s 2 k= α v k/2 (k 2/2 u Φ k (s y k F (ydy 8C(εC 2 sup{z 2 F (z; z }α s 7+ε Φ (s exp(α s. Sice sup sup s>β s 7+ε Φ (s exp(α s <, wehave sup s>β ( F (u s I (, s Therefore we have our assertio. 2 k= α u v k/2 (k 2/2 Φ k (s y k F (ydy,. Propositio 5. sup H (, s I 2 (, s s [, 2 k= v k/2 (2 k/2 Φ k (s ( 7/8u s y k F (ydy,. α u
48 Keji Nakahara Proof. Similarly to Propositio 4, we see that 2 ( 7/8u I 2 (, s v k/2 (2 k/2 s Φ k (s y k F (ydy k= α u ( 7/8u u s F (y (, s, y dy α u ( u 3 F 7/8u (u α C ( + s 2 s ( sup Φ (z y 2 dy z [ 7/8s,s] u α 4C F (u α s 5 Φ ( 7/8s 4C +7/8 F (u α s 6 Φ (s 7/8. Sice H (, s Φ (s 6/7 ( F (u s /7,itiseasy to see that for ay ε (, 4/7, there is a costat C > such that 2 ( 7/8u H (, s I 2 (, s v k/2 (2 k/2 s Φ k (s y k F (ydy k= C s 6+2/7+ε Φ (s 7/8 6/7 L(u ( ε/3 /7. α u Sice sup s {s 6+2/7+ε Φ (s 7/8 6/7 } < ( ε/3 /7 >, we have sup H (, s I 2 (, s s Propositio 6. 2 k= v k/2 (2 k/2 Φ k (s ( 7/8u s y k F (ydy,. α u sup H (, s I 3 (, s F (u s, s [,. Proof. I 3 (, s = F 7/8s (u s s s F (u (s x Φ (xdx F (u s = F 7/8s (u s ( x L(u (s x s 2 Φ (xdx. L(u s
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 49 It is easy to see that there is a costat C > such that Φ (s C L(u 2/3,,s [, ( log L(u /2 ], 7/8s F (u (s x Φ (xdx C,,s [,. F (u s s The we have sup H (, s I 3 (, s F (u s s ( log L(u /2 C (C +L(u 2/3 F (u C (C +v L(u /3,. We take M> arbitrarily, the ( log L(u /4 >M for sufficietly large. Hece we see that for s>( log L(u /2 7/8s ( x L(u (s x s s 2 Φ (xdx L(u s 7/8s {( x s s 2 } L(u (s x Φ (xdx L(u s 7/8s + ( L(u (s x Φ (xdx s L(u s + Φ (xdx [ s, 7/8s] C ( M 2 ( x s 2 Φ (xdx +8Φ (M M + sup sup t>( log L(u /2 7/8 a L(at L(t +2Φ ( 7/8s. Hece we have sup F (u s I 3 (, s F (u s, s>( log L(u /2. So we have our assertio.
42 Keji Nakahara Propositio 7. sup s [, I 4 (, s H (, s,. Proof. I 4 (, s F (2u sφ (s. Hece we have Φ (s I 4 (, s F (2u s F (u,. Propositio 8. sup s [, H (, s v /2 /2 Φ (s 7/8u s F (ydy, sup H (, s s [, ( /2 v Φ 2 (s y F (ydy + L( /2, 7/8u s. Proof. From Propositio 3 (2, we see that there is a costat C > such that /2 ( 7/8u s F (ydy C s L(( 7/8u s. We ca easily see that sup Φ (s /2 Φ (s s [,β ( 7/8 /2 s F (ydy, sup ( F ( /2 s /2 Φ (s s [β, ( 7/8 /2 s F (ydy,.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 42 Also we see that for ay ε (,, there is a costat C 2 > such that /2 ( 7/8u s y F (ydy = Hece we ca easily see that ( 7/8v /2 s L( /2 x dx C 2 L( /2 s ε. x sup s [,β Φ (s ( /2 Φ 2 (s y F (ydy + L( /2 ( 7/8u s, sup s [β, ( F ( /2 s ( /2 Φ 2 (s y F (ydy + L( /2 ( 7/8u s,. Therefore we have our assertio. Now let us prove Lemma 2. Note that H(, s H (, s =A(, s F (s /2. So Propositios 4, 5, 6, 7 8 imply Lemma 2. 7. Proof of Theorem 2 4 First we prove the followig lemma. Lemma 3. For ay β> δ (,, there is a costat C> such that sup P ( k= X k >s /2 s>l( /2 β H(, v /2 CL( /2 δ. s We make some preparatio to prove Lemma 3. Similarly to Propositio 26 i Fushiya-Kusuoka [2], we ca prove the followig.
422 Keji Nakahara Propositio 9. ( For ay t, s >, 2, P ( X k {Xk t /2 } >s/2 exp( 6 t 2 s t. k=2 (2 For ay s, t >, ε (, with t<( εs, P ( X k >s /2 P (X + k= X k {Xk t /2 } >s/2, k=2 X k {Xk t /2 } εs/2 k=2 2( F (t /2 2 + exp( 6 t 2 s t + F (t /2 exp( 6 t 2 εs 2t. Also we prove the followig for the proof of Lemma 3. Propositio 2. For ay γ, δ, ε (, β>, there is a costat C> such that P (X + X k {Xk s γ /2 } >s/2, X k {Xk s γ /2 } εs/2 k=2 /2 εv s k=2 F ( /2 (s v /2 xφ (xdx C F (( ε /2 sl( /2 3δ, for s>l( /2 β. Proof. It is easy to see that there is a costat C > such that P (X + X k {Xk s γ /2 } >s/2, X k {Xk s γ /2 } εs/2 k=2 P (X + X k >s /2, k=2 k=2 k=2 X k εs /2, X 2 L( /2 δ /2,...,X L( /2 δ /2 C F (( ε /2 sl( /2 3δ, for s>l( /2 β.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 423 We see that P (X + X k >s /2, k=2 =( F (t εs X k εs /2,X 2 t,...,x t k=2 F ( /2 (s xµ(t ( (dx, here t = L( /2 δ /2. Similarly to the proof of Propositio 2, we have our assertio. Now let us prove Lemma 3. Sice H(, v /2 s = Φ (v /2 s+ /2 εv s /2 v s εv /2 s v /2 /2 Φ (v /2 s = Φ (v /2 s v F ( /2 (s v /2 xφ (xdx F ( /2 (s v /2 xφ (xdx Φ 2 (v /2 s 2 xµ(dx v /2 x 2 µ(dx Φ 2 (v /2 s /2 x 2 µ(dx 2 + v /2 /2 η (( ε /2 sφ (v /2 s + v /2 /2 ( ( ε /2 s F (z(φ (v /2 s /2 v /2 z Φ (v /2 sdz, it is easy to see that there is a costat C > such that H(, v /2 s /2 εv s C sφ (εv /2 s, fors. F ( /2 (s v /2 xφ (xdx Combiig Propositio 9 (2 2, we see that there is a costat C 2 >
424 Keji Nakahara such that P ( X k >s /2 k= /2 εv s F ( /2 (s v /2 xφ (xdx 2( F (s γ /2 2 + exp( 6 s 2γ s s γ + F (s γ /2 exp( 6 εs s2γ 2s γ + C 2 F (( ε /2 sl( /2 δ. Hece we see that there is a costat C> such that sup ( F ( /2 s P ( s>l( /2 β CL( /2 δ. k= X k >s /2 H(, v /2 s Therefore by Lemma 2, we have our assertio. Now let us prove Theorem 4. By Theorem 2, we see that there is a costat C > such that P ( k= X k >s /2 H(, v /2 s C L( /2 2 δ/2, s. Note that for ay ε>, there is a costat C 2 > such that F ( /2 s C2 s 3 L( /2 C2 L(/2 +δ/2 for s L( /2 δ/6. Hece by Lemma 2, we see that there is a costat C 3 > such that So we have H(, v /2 s C 3 ( F ( /2 s C 2 C 3 L( /2 +δ/2, s L( /2 δ/6. sup P ( k= X k >s /2 s L( /2 δ/6 H(, v /2 C C 2 C 3 L( /2 δ. s From this Lemma 3, we have Theorem 4. Theorem 2 is a easy cosequece of Theorem 4 Lemma 2.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 425 8. Proof of Theorem 3 First let us assume lim sup ( v log L( /2 that Φ (s Φ (v /2 s = Let z = have /2 v s s Φ (zdz = v s 2 s 2 =. The we see e y/2 dy 2π 2 y s ( v Φ (s 2v s 2 C ( v Φ (s. 2v L( /2, the we have lim sup ( v log z =. Hece we sup s [, Φ (v /2 s+ F ( /2 s 3 log z Φ (s+ F ( /2 s 3C 2v ( v log z,. We also see that for s> 3 log z, Φ (v /2 s+ F ( /2 s Φ (s+ F ( /2 C ( v s 2 Φ (s s 2v Φ (s+ F ( /2 s C ( v s 4 Φ (s 2v L( /2 C2 s 2 s 5 exp( s 2 /2 L(/2 2πv L( /2 s z C 2 2 s 6 exp( s 2 /2z 2πv C 2 2 sup 2πv s s 6 exp( s 2 /6, 3 log z. Hece we have sup s [, Φ (v /2 s+ F ( /2 s Φ (s+ F ( /2,. s Next, we assume lim sup ( v log L( /2 >. Let y =(
426 Keji Nakahara v log z s = log z. The lim sup y >. Hece we see that lim if Φ (s Φ (v /2 lim if exp( v So we have s =lim if v/2 Φ (s Φ (v /2 s ( v s 2 =exp( lim sup y < Φ (s F ( /2 s C s Φ (s s 2 L( /2 s 2πC M(L( /2 /2,. lim if Φ (v /2 s + F ( /2 s Φ (s + F ( /2 <. s Therefore we have Theorem 3. We give a example i the rest of this sectio. Let x L : [x, (, beac 2 slowly varyig fuctio satisfyig L(x dx <, L(x,x, x x sup ( L (x + L (x <. x x The we ca fid F : [, ] o-decreasig C 2 fuctio with F ( =, F ( =, F (x dx < F (x =x 2 L(x for sufficiet large x >. Let µ be a probability measure whose distributio fuctio is F. The we see that µ satisfies (A3. Let L(x =(log x (log log x b, b> for sufficietly large x>. We ca easily see that L(x satisfies the above coditios. For sufficietly large, we see that v = /2 x 2 µ(dx =L( /2 +2 L(x /2 x dx = L( /2 + 2 (log log log 2 b b 2 b (log log b. Hece we have the followig.
Sum of i.i.d. om Variables with Fat Tail i the Threshold Case 427 Propositio 2. Let L(x =(log x (log log x b, b> for sufficietly large x>. The we have lim sup( v log L( /2 >, for b (, ] lim ( v log L( /2 =, for b (,. Therefore ( does ot hold for b (, ]. efereces [] Borovlov, A. K. Borovkov, Asymptotic Aalysis of om Walks: Heavy Tailed Distributios, Cambridge Uiversity Press 28, Cambridge. [2] Fushiya, H. S. Kusuoka, Uiform Estimate for Distributios of the Sum of i.i.d. om Variables with Fat Tail, J. Math. Sci. Uiv. Tokyo 7 (2, 79 2. [3] Nagaev, A. V., Itegral limit theorems takig large deviatios ito accout whe Cramér s coditio does ot hold. I,II, Theor. Probab. Appl. 4 (969, I,II. 5 64, 93 28. 745 789. [4] Nagaev, A. V., Limit theorems that take ito accout large deviatios whe Cramér s coditio is violated, Izv. Akad. Nauk UzSS Ser. Fiz-Mat. Nauk 3 (969, o. 6, 7 22 (I ussia. 745 789. [5] Nagaev, S. V., Large deviatios of sums of idepedet rom variables, A. Probab. 7 (979, 745 789. [6] ozovskii, L. V., Probabilities of large deviatios o the whole axis, Theory Probab. Appl. 38 (993, 53 79. (eceived December 25, 2 (evised October 2, 2 Kuriya 4-5-6, Tama-ku Kawasaki-shi Kaagawa 24-39, Japa E-mail: k.akahara9@gmail.com