Main result

3.5. Remainder term estimate: 0< α <1 61

= 1 +ϕ_α,1(t) + ϕ²_α,1(t)

2! + ϕ³_α,1(t) 3! +. . .

!

1 +

s

X

j=2

A_jϕ^j_α,1(t)





= 1 +ϕ_α,1(t) +

1 2+A₂

ϕ²_α,1(t) +

1

6+A₂+A₃

ϕ³_α,1(t) +. . .

+ 1

m! +

m

X

u=2

A_u (m−u)!

!

ϕ^m_α,1(t) +· · ·+ 1 s!+

s

X

u=2

A_u (s−u)!

!

ϕ^s_α,1(t) +

∞

X

k=s+1

ϕ^k_α,1(t) 1 k!+

s

X

u=2

A_u (k−u)!

!

= 1 +ϕ_α,1(t) +

s

X

k=2

ϕ^k_α,1(t) 1 k! +

k

X

u=2

A_u (k−u)!

!

+

∞

X

k=s+1

ϕ^k_α,1(t) 1 k!+

s

X

u=2

A_u (k−u)!

!

= 1 +ϕ_α,1(t) +

∞

X

k=s+1

ϕ^k_α,1(t) 1 k! +

s

X

u=2

A_u (k−u)!

!

. Thus,

f(t)−g(t) =_e

∞

X

n=1

c^n/α₁ α α−n

(it)ⁿ n! −

∞

X

k=s+1

ϕ^k_α,1(t) 1 k!+

s

X

u=2

A_u (k−u)!

!

.

From this expansion for fixed s we can obtain formulas for pseudomoments µ^∗_j. Indeed, we have

f(t)−g(t) =e

[(1+s)α]

X

n=1

c^n/α₁ α α−n

(it)ⁿ

n! +o|t|^α(s+1) as t→0, whence from Lemma 3.25 it follows that

µ^∗_j =c^j/α₁ α

α−j for j = 1, . . . ,[(1 +s)α].

It is important to note that s can be chosen arbitrarily. This means that for the Pareto distribution we can always construct such H^f that we have as many finite pseudomomentsµ^∗_j :=µ_j(H) as we want.^f

c_i :=C_i(α), i= 1, ..., s, from (58) and d_i are suitable constants for i= 2, . . . , s.

Our goal is to build an approximation of the distribution function F_n defined by F_n(x) :=P

Sn−an

b_n ≤x

,

whereS_n=X₁+X₂+· · ·+X_nand (a_n), (b_n) are some suitably chosen normalizing sequences such that F_n →G_α,1 as n → ∞. From the definition of DNA(G_α,1) (see Definition 3.5) it follows thatb_n=a n^1/α with a >0. Without loss of generality we put a= 1, and also for α ∈(0,1) we can take a_n= 0, n ∈N. This means, we have

F_n(x) = P X₁+· · ·+X_n≤xn^1/α=F^n∗(xn^1/α).

It is known thatF_n→G_α,1, but we want to construct a correction term forG_α,1(x) in order to get a better approximation. Just as in Theorem 3.10 or Theorem 3.21 this correction function is a linear combination of derivatives of the corresponding limit distribution. The only difference is that the coefficients of this linear combination depend on new pseudomoments µ^∗_k =µ_k(F −G^e_α) defined by (79).

For a given distribution functionF we construct a functionG^e_αusing formula (69) and fix it. In terms of new pseudomoments µ^∗_k for some r ∈ R+, r > α and n ∈N we construct the function W^f_r,n(x) for all x∈R as follows:

Wf_r,n(x) =

ρ

X

k=2

c_k,n

n^k G^(0,k)(x,1) (87)

+

p

X

k=0 mk

X

`=1

n

`

!ck,n−`

n^k

m_`,k

X

u=`

p_u,`,k

X

v=0

G^(u,k+v)(x,1)(−`/n)^v(−1)^u

v! n^−u/α C^e_{u,`</sub>, where ρ = [2(R + 1)/α], p = [2R/α], m<sub>`,k} = [R + 1 + α(` − 1 − k/2)], m<sub>k</sub> = 1 + [(R−αk/2)/(1−α)], p<sub>u,`,k</sub> = max{0,[(R+ 1−u)/α+`−1−k/2]}

with

R =

( [r], if [r]6=r

r−1, if [r] =r , (88)

c_{r, ρ}= ^X

k0+k2+···+ks=ρ 2k2+···+sks=r

ρ!

k₀!k₂!· · ·k_s!A^k₂²· · ·A^k_s^s, r, ρ∈N0, (89) s is from (85), Aj,j = 2, ..., s, are from (70),G^(u,k)(x,1) are defined by (71) and

Ceu,` = ^X

k1+2k2+...+RkR=u k1+k2+...+kR=`

`!

k₁!...k_R!

µ^∗₁ 1!

^k1

...

µ^∗_R R!

^kR

, u=`, ..., m`,k. (90)

Then we have the following analogue of Theorem 3.21.

Theorem 3.26. If for r > 1 we have 0 < γ_r^∗ < ∞ and ν₀^∗ <1, then for all x ∈R and all integersn ≥2 the following inequality holds:

Fn(x)−Gα,1(x)−W^fr,n(x)≤C(1 +|x|)^−rn^−r−αα 1 +n^αr Qn

,

3.5. Remainder term estimate: 0< α <1 63

where constant C does not depend on x and n, W^f_r,n(x) is defined by (87) and Q_n =ν₀^∗n−1+sup_|t|>

e^ε

|f(t)|+ 2γ_r^∗n^−r/αn−1 with ε_e defined as follows:

εe= min







1, 1

c₀, 1

(2D)^1/α, 1 D^1/α





cos^απ₂ 8D





2+ρ/2 α

,





cos^απ₂ 16e c₀





1/(1−α)





, (91)

where c₀ = (ν₀^∗ + 1)γ_r^∗1/r, D = max

2≤j≤s{2|A_j|^1/j}, ρ = [2(R+ 1)/α] with A_j defined by (70) and R defined by (88).

Proof. Section 4 will be devoted to the proof of this result.

Remark 3.34. If ν₀^∗ < 1, then for each ε > 0 we have sup_|t|>ε|f(t)| < 1. Indeed, from formula (77) for eg_α(t) and from definition (80) ofν₀^∗ we have

|f(t)−g_eα(t)|=

Z +∞

−∞

e^itxdF −G^e_α(x)

≤

Z +∞

0

dF −G^e_α(x)=ν₀^∗. Then, using formulas (77), (74) forge_α(t) and the fact thatν₀^∗ <1 we obtain:

|f(t)| ≤ν₀^∗+|ge_α(t)| ≤ν₀^∗+|g_α,1(t)|



1 +

s

X

j=2

|A_j| |ϕ^j_α,1(t)|





≤ν₀^∗+e^{−|t|αcos(α π/2)}



1 +

s

X

j=2

|A_j| |t|^{α j}



<1 for large |t|.

Finally, from Lemma A.16 it follows that sup_|t|>ε|f(t)|<1 for each ε >0.

Remark 3.35. Note that there exists such n₀ ∈ N that n^r/αQ_n ≤ 1 for all n ≥ n₀, since ν₀^∗ <1 and, as a result, sup_|t|>

e^ε

|f(t)|<1 (see Remark 3.34).

Example 3.11. Let us consider a Pareto-distributed random variable X with α = 1/2 and κ = c²₁, where c₁ = C₁(1/2) = 1/√

π is defined in (58). The dis-tribution functionF of X has the form:

1−F(x) = 1

√π√

x, x≥1/π.

Comparing this representation with representation (85) we can make the following conclusions: s can be chosen equal to 3 (since 3α ≥ 1 +α), u(x) = 0 and d₃ = 0.

Coefficient d2 can be chosen arbitrarily, since c2 =C2(1/2) = 0 (see formula (58)).

Let us put d₂ = 3/4.

In order to apply Theorem 3.26 we have to check the condition ν₀^∗ < 1 and to decide for which r we have γ_r^∗ <∞ and ν_r^∗ < ∞. According to the definition (80) we have

ν_r^∗ =

Z ∞ 0

x^rdF −G^e_1/2(x), r ≥0, where

Ge_1/2(x) =G_1/2,1(x) +

3

X

j=2

A_jG^(0,j)(x,1)

with coefficients A₂ =−1/8 and A₃ =−1/24 (see (69) and (70)), chosen in such a way, that

F(x)−G^e_1/2(x) =Ox^−5/2 as x→ ∞.

Note that we obtainO(x^−5/2) and not just O(x^−(s+1)α) = O(x⁻²), sinceC4(1/2) = 0 in (57). Therefore, γ_5/2^∗ <∞ and ν_r^∗ <∞ for r <5/2. So, we apply Theorem 3.26 with r= 5/2.

Let us check whether the condition ν₀^∗ < 1 is satisfied. Note that the stable distribution corresponding toG_1/2,1(x, λ) is a Lévy distribution which has an explicit density function

p_1/2,1(x, λ) = λ 2√

πe^−λ

2

4xx⁻³², x >0, λ >0.

Using this we get ν₀^∗ =

Z ∞ 0

dF −G^e_1/2(x)=

Z ∞ 0

p(x)−pe_1/2(x)dx, where

pe_1/2(x) = 1 384√

π e^−41x x⁻⁹² 192x³+ 48x²−18x+ 1.

Using software Mathematica we obtain ν₀^∗ ≈ 0.32 < 1. Thus, Theorem 3.26 is applicable. Now, let us see now how the correction termW^f_5/2,n(x) looks like in this particular case.

Wf_5/2,n(x) =

12

X

k=2

c_k,n

n^k G^(0,k)(x,1) +

8

X

k=0

1+[4−k/2]

X

`=1

n

`

!ck,n−`

n^k

m`,k

X

u=`

pu,`,k

X

v=0

G^(u,k+v)(x,1)(−`/n)^v(−1)^u

v! n^−u/α C^e_{u,`</sub>, wherem<sub>`,k} =^h3 + ¹₂(`−1− <sup>k</sup><sub>2</sub>)<sup>i</sup>, p<sub>u,`,k</sub> = maxⁿ0,^h2 (3−u) +`−1− ^k₂^ioand

c_{r, ρ} = ^X

k0+k2+k3=ρ 2k2+3k3=r

ρ!

k₀!k₂!k₃!

−1 8

k2 −1 24

k3

, r, ρ∈N0,

Ce_u,` = ^X

k1+2k2=u k1+k2=`

`!

k₁!k₂!

µ^∗₁ 1!

k1µ^∗₂ 2!

k2

, u=`, ..., m<sub>`,k</sub>.

The first and the second pseudomoments µ^∗₁, µ^∗₂ can be found precisely using the same method as in Example 3.10. The only difference is that we do not putd₂ equal to 0. We get

µ^∗₁ =

Z ∞ 0

x dF −G^e_1/2(x) = c^1/α₁ α α−1 +

1 2+A₂

= 3 8 − 1

π, µ^∗₂ =

Z ∞ 0

x²dF −G^e_1/2(x) = 2!





c^2/α₁ α 2(α−2)−

1 24+1

2A₂+A₃



= 1 8− 1

3π².

3.5. Remainder term estimate: 0< α <1 65

From Theorem 3.26 it follows that for allx∈R and n ≥2 we have

F_n(x)−G_1/2,1(x)−W^f_5/2,n(x)≤C(1 +|x|)^−5/2n⁻⁴ 1 +n⁵Q_n, whereQ_n =ν₀^∗n−1+sup_|t|>

eε|f(t)|+ 2γ_r^∗n^{−5/(2α)n−1} with εedefined by (91).

4 Proof of the main result

4.1 Some auxiliary functions and plan of the proof

In order to prove Theorem 3.26 we need to introduce some auxiliary functions. The first one is a truncated distribution function. We define it for any fixed n ∈N and ξ∈[0,∞) as follows:

F_n,ξ(y) =

( F(y) for y≤n^1/α(1 +ξ),

Ge_α(y) for y > n^1/α(1 +ξ), y ∈R. (92) Thus, we have a family of truncated functions: Fn,ξ

n∈N,ξ∈[0,∞). We denote H_n,ξ(x) := F_n,ξ(x)−G^e_α(x), i.e.

H_n,ξ(x) =

(

H(x) =f F(x)−G^e_α(x) for x≤n^1/α(1 +ξ),

0 for x > n^1/α(1 +ξ), x∈R, (93) and consider pseudomoments µ_i,n,ξ = µ_iH_n,ξ and ν_r,n,ξ = ν_rH_n,ξ. They are well-defined for the same reasons as those for whichµ^∗_i and ν_r^∗ are well-defined (see Remark 3.33). Moreover, for any i ∈ N⁰ and any r ≥ 0 we have |µ_i,n,ξ| < ∞ and ν_r,n,ξ <∞. Indeed, puttingN =n^1/α(1 +ξ), taking into account the possible jump of H(x) at point x = N, and using the boundedness of ν₀^∗ (see Lemma 3.24 (ii)) and of G^e_α (see (72)), we obtain

ν_r,n,ξ = ν_rH_n,ξ=

Z +∞

−∞

|x|^r dH_n,ξ(x)=

Z N 0

x^r dH(x)^f +N^rH(N^f )

≤ N^r

Z N 0

dH(x)^f +H(N^f )

!

≤N^rν₀^∗+F(N)−G^e_α(N)

≤ N^rν₀^∗ + 1 +G^e_α(N)<∞ ∀r≥0. (94) From (94) and from Lemma 3.17 it follows that|µ_i,n,ξ| ≤ν_i,n,ξ <∞ for any i∈N0. Note also that for alln ∈N and ξ≥0 we have

F_n,ξ(x) =H_n,ξ(x) = 0 for x <0. (95) This follows from Lemma 3.22 (i) and from the fact that we consider only non-negative random variables, i.e. we haveF(x) = 0 for x <0.

Let us consider one more function. We denote M_n,ξ(x) := H_n,ξ(x)−H_n,0(x).

Using the definition of H_n,ξ we obtain M_n,ξ(x) =

( F(x)−G^e_α(x), if x∈n^1/α, n^1/α(1 +ξ)ⁱ,

0, otherwise, x∈R. (96)

Note thatM_n,ξ and H_n,ξ are functions of bounded variation. This follows from the fact thatH(x) =^f F(x)−G^e_α(x) is a function of bounded variation (see Remark 3.33).

Thus, pseudomoments µk

Mn,ξ

, k ∈N⁰, and absolute pseudomoments νr

Mn,ξ

,

r ≥ 0, are well-defined. From the definition of H_n,ξ, M_n,ξ, Lemma 3.17 and in-equality (94) it follows that

µ_i(M_n,ξ)≤ν_i(M_n,ξ)≤ν_i(H_n,ξ) = ν_i,n,ξ <∞, i∈N0. (97) From Lemma 3.22 (iii) and from the definition of Mn,ξ it follows that

x→+∞lim M_n,ξ(x) = 0 for all n∈N and ξ≥0. (98) The following two lemmata give some properties of pseudomoments µ_i,n,ξ, µ_i(M_n,ξ) and ν_r,n,ξ as well as the connection between them and pseudomoments µ^∗_i and ν_r^∗. Recall that for any r∈R⁺ we denote

R =

( [r], if [r]6=r,

r−1, if [r] =r. (99)

Lemma 4.1. For pseudomomentsµ_i,n,ξ, µ_iM_n,ξand µ^∗_i the following statements hold true.

(i) µ_0,n,ξ =µ₀M_n,ξ= 0 for all n∈N and ξ≥0;

(ii) If for r >1 we have 0< γ_r^∗ <∞, then

µ_u,n,ξ−µ^∗_u≤2 (n^1/α(1 +ξ))^u−rγ_r^∗, u= 1, . . . , R, where R is given by (99).

Proof. (i)We put N =n^1/α(1 +ξ). Then, µ_0,n,ξ =

Z +∞

−∞ dH_n,ξ(x) =

Z +∞

0

dH_n,ξ(x) =

Z +∞

0

dF_n,ξ(x)−

Z +∞

0

dG^e_α(x)

=

Z N 0

dF(x) +

Z +∞

N

dG^eα(x) +G^eα(N)−F(N)−

Z +∞

0

dG^eα(x)

=

Z N 0

dF(x)−F(N) = 0.

Using the last equality and the definition of M_n,ξ(x) =H_n,ξ(x)−H_n,0(x) we obtain µ0

Mn,ξ

=

Z +∞

−∞ dMn,ξ(x) =

Z +∞

−∞ dHn,ξ(x)−

Z +∞

−∞ dHn,0(x) = µ_0,n,ξ−µ_0,n,0 = 0.

(ii) Using the fact that lim

x→+∞H(x) = 0, which follows from equality (73), we havef

foru= 1, . . . , R:

µ_u,n,ξ−µ^∗_u =

Z +∞

0

y^udH_n,ξ−H^f(y)

=

Z +∞

N

y^udH(y) +^f N^uH(N^f )

≤

Z +∞

N

y^udH(y)^f +

N^u

Z +∞

N

dH(y)^f

4.1. Some auxiliary functions and plan of the proof 69

≤

Z +∞

N

y^udH(y)^f +N^u

Z +∞

N

dH(y)^f

≤ 2

Z +∞

N

y^udH(y)^f .

We distinguish two cases. For [r]6=r we obtain

µ_u,n,ξ−µ^∗_u≤2

Z +∞

N

y^udH(y)^f ≤2N^u−rN^r−[r]

Z +∞

N

y^[r]dH(y)^f ≤2N^u−rγ_r^∗. For [r] =r we have

µ_u,n,ξ−µ^∗_u ≤ 2

Z +∞

N

y^udH(y)^f ≤2N^u−rN

Z +∞

N

y^r−1dH(y)^f

≤ 2N^u−r

Z N

0

x^rdH(x)^f

+N

Z +∞

N

y^r−1dH(y)^f

!

≤ 2N^u−rγ_r^∗. This completes the proof of the lemma.

Lemma 4.2. If for r >1 we have 0< γ_r^∗ <∞, then (i)

ν_k,n,ξ ≤ν_k^∗ ≤(ν₀^∗+ 1)γ_r^∗k/r, k = 1, . . . , R, where R is defined by (99).

(ii)

ν_q,n,ξ ≤Cn^1/α(1 +ξ)^q−rγ_r^∗, q > r, q ∈R, (100) where C is some constant, which depends only on q and r.

Proof. (i) As always we put N =n^1/α(1 +ξ). Using the fact that lim

x→+∞H(x) = 0f

we have for k= 1, . . . , R:

ν_k,n,ξ =

Z +∞

−∞ x^kdH_n,ξ(x)=

Z +∞

0

x^kdH_n,ξ(x)=

Z N 0

x^kdH(x)^f +N^kH(N^f )

=

Z N 0

x^kdH(x)^f +N^k

Z +∞

N

dH(x)^f

≤

Z N 0

x^kdH(x)^f +

Z +∞

N

x^kdH(x)^f =ν_k^∗.

The other inequality from(i) we obtain as follows with z =γ_r^∗1/r: ν_k^∗ =

Z +∞

0

x^kdH(x)^f =

Z z 0

x^kdH(x)^f +

Z +∞

z

x^kdH(x)^f

≤ z^k

Z z 0

dH(x)^f +











z^k−rz^r−[r]

Z +∞

z

x^[r]dH(x)^f , if [r]6=r, z^k−rz

Z +∞

z

x^r−1dH(x)^f , if [r] =r

≤ z^kν₀^∗+z^k−rγ_r^∗ ≤(ν₀^∗+ 1)γ_r^∗k/r.

(ii) We denote T(x) = −^R_x^+∞z^RdH(z)^f . Using integration by parts, we obtain νq,n,ξ =

Z +∞

−∞ x^qdHn,ξ(x)=

Z +∞

0

x^qdHn,ξ(x)=

Z N 0

x^q−RdT(x) +N^qH(N^f )

= x^q−RT(x)^x=N

x=0 −

Z N 0

(q−R)x^q−R−1T(x)dx+N^q−R

Z +∞

N

x^RdH(x)^f

≤ N^q−RT(N) +

Z N 0

(q−R)x^q−R−1

Z +∞

x

z^RdH(z)^f dx−N^q−RT(N)

= (q−R)·









 Z N

0

x^q−r−1x^r−[r]

Z +∞

x

z^[r]dH(z)^f dx, if r6= [r],

Z N 0

x^q−r−1x

Z +∞

x

z^r−1dH(z)^f dx, if r = [r]

≤ (q−R)γ_r^∗

Z N 0

x^q−r−1dx≤ q−R

q−r N^q−rγ_r^∗. The lemma is proved.

The following lemma gives us the estimation of absolute pseudomoments ofH^`∗_n,ξ in terms of pseudomoments ofH_n,ξ.

Lemma 4.3. For all n, `∈N, ξ≥0 and all q∈N0 we have νq

H^`∗_n,ξ

≤`<sup>q</sup> νq,n,ξ ν<sup>`−1</sup>_0,n,ξ <∞. (101) In particular, the absolute pseudomoment νq

H^`∗_n,ξ

is finite.

Proof. Using (95), the definition of the `-fold convolution of H_n,ξ, inequality (148) from Lemma A.6 and the fact thatν_r,n,ξ is finite for anyr ≥0 (see (94)), we obtain forq ∈N:

ν_q

H^`∗_n,ξ

=

Z +∞

0

x^q

dH^`∗_n,ξ(x)

=

Z +∞

0

x^q

d

Z +∞

0

H^(`−1)∗_n,ξ (x−y₁)dH_n,ξ(y₁)

≤

Z +∞

0

. . .

Z +∞

0

| {z }

`

(y₁+· · ·+y_{`</sub>)<sup>q</sup>dH<sub>n,ξ</sub>(y<sub>1</sub>). . .dH<sub>n,ξ</sub>(y<sub>`})

≤ `^q

Z +∞

0

dH_n,ξ(x)

`−1Z +∞

0

x^qdH_n,ξ(x)≤`<sup>q</sup> ν<sub>q,n,ξ</sub>ν<sup>`−1</sup>_0,n,ξ <∞.

Similarly we can show that ν₀

H^`∗_n,ξ

≤ν^`_0,n,ξ. The lemma is proved.

Below we give one more useful lemma about the pseudomoments of H^`∗_n,ξ. Lemma 4.4. The following properties hold true.

(i) For all u, `∈N we have µ_u

H^`∗_n,ξ

= ^X

k1+k2+···+k_`=u

u!

k₁!· · ·k_{`</sub>!µ<sub>k1,n,ξ</sub>· · ·µ<sub>k`,n,ξ}.

4.1. Some auxiliary functions and plan of the proof 71

(ii) For all u, `∈N with u < ` we have µ_u

H^`∗_n,ξ

= 0.

(iii) For fixed R≥1, ` ∈N and u= 1, ..., R we have µ_u

H^`∗_n,ξ

=u! ^X

k1+k2+···+k_R=`

k1+2k2+...+RkR=u

`!

k1!· · ·kR!

µ_1,n,ξ 1!

!k1

· · · µ_R,n,ξ R!

!kR

.

Proof. (i) Using (95) and rewritingµ_u

H^`∗_n,ξ

and using formula (147) from Lemma A.5 we obtain

µ_u

H^`∗_n,ξ

=

Z +∞

0

y^udH^`∗_n,ξ(y)

=

Z +∞

0

...

Z +∞

0

| {z }

`

(z₁+· · ·+z_{`</sub>)<sup>u</sup>dH<sub>n,ξ</sub>(z<sub>1</sub>)· · ·dH<sub>n,ξ</sub>(z<sub>`})

=

Z +∞

0

...

Z +∞

0

| {z }

`

X

k1+k2+···+k_`=u

u!

k₁!· · ·k_{`</sub>!z<sub>1</sub><sup>k1</sup>· · ·z<sub>`}<sup>k`</sup>dHn,ξ(z<sub>1</sub>)...dH<sub>n,ξ</sub>(z<sub>`</sub>)

= ^X

k1+k2+···+k_`=u

u!

k₁!· · ·k_`! µ_k1,n,ξ· · ·µ_k

`,n,ξ.

(ii) If u < `, then there exists such i ∈ {1, . . . , `} that k_i = 0. Therefore each productµ_k1,n,ξ· · ·µ_k`,n,ξ from above containsµ_0,n,ξ. According to Lemma 4.1 (i) we haveµ_0,n,ξ = 0. This impliesµ_u

H^`∗_n,ξ

= 0 for u < `.

(iii) Note that if u < `, then the right side of the equality in (iii) is equal to 0 since we have to sum up over the empty set. This fact together with statement (ii) gives us the statement of (iii) foru < `. Let us consider now u≥`. Using the properties of the inverse Fourier transformh_n,ξ(t) ofH_n,ξ(x) (similarly to Lemma 3.25) we find

h_n,ξ(t) :=

Z +∞

−∞

e^itxdH_n,ξ(x) = it

1!µ_1,n,ξ+· · ·+(it)^R

R! µ_R,n,ξ+O(|t|^R+1), t→0.

Using this expansion and formula (147) from Lemma A.5 we obtain fort →0:

h^`_n,ξ(t) = it

1!µ_1,n,ξ +...+(it)^R R! µ_R,n,ξ

!`

+O(|t|^R+1)

= ^X

k1+...+kR=`

`!

k₁!...k_R!

µ1,n,ξ

1!

k1

...

µR,n,ξ

R!

kR

(it)^{k1+2k2+...+RkR} +O(|t|^R+1)

=

R

X

u=`

(it)^u

u! u! ^X

k1+k2+···+kR=`

k1+2k2+...+RkR=u

`!

k₁!· · ·k_R!

µ_1,n,ξ 1!

!k1

· · · µ_R,n,ξ R!

!kR

+O(|t|^R+1).

On the other hand we can considerh^{`</sup><sub>n,ξ</sub>(t) as the inverse Fourier transform ofH<sup>`∗}_n,ξ(x), i.e. we have

h^`_n,ξ(t) = it 1!µ₁

H^`∗_n,ξ

+· · ·+(it)^R R! µ_R

H^`∗_n,ξ

+O(|t|^R+1), t→0.

Taking into account property (ii) and comparing coefficients in both representations of h^`_n,ξ we get the statement of (iii). The lemma is proved.

We need one more auxiliary function, which is similar to the function W^f_r,n from (87). Forr ∈R+, r >1, n ∈N and ξ∈[0,∞) we define

Wr,n,ξ(x) =

ρ

X

k=2

c_k,n

n^k G^(0,k)(x,1) +W^∗_n,ξ(x) +

p

X

k=0 mk

X

`=1

n

`

!ck,n−`

n^k

m`,k

X

u=`

pu,`,k

X

v=0

G^(u,k+v)(x,1)(−`/n)^v(−1)^u

v! n^−u/α C_u,`,

(102)

where ρ = [2(R + 1)/α], p = [2R/α], m<sub>`,k</sub> = [R + 1 + α(` − 1 − k/2)], m_k = 1 + [(R−αk/2)/(1−α)], p<sub>u,`,k</sub> = max{0,[(R+ 1−u)/α+`−1−k/2]}

with

R =

( [r], if [r]6=r r−1, if [r] =r , c_{r, ρ}= ^X

k0+k2+···+k_s=ρ 2k2+···+sks=r

ρ!

k₀!k₂!· · ·k_s!A^k₂²· · ·A^k_s^s, r, ρ∈N0, (103) s and Aj,j = 2, ..., s, are from (86) and (70), respectively,

C_u,` = ^X

k1+2k2+...+RkR=u k1+k2+...+kR=`

`!

k₁!...k_R!

µ_1,n,ξ 1!

!k1

... µ_R,n,ξ R!

!kR

, (104)

and

W^∗_n,ξ(x) =n G^(R+1,0)(x,1)(−1)^R+1

(R+ 1)! n^−R+1α µ_R+1,n,ξ +n G_α,1(·, n)∗M_n,ξ(xn^1/α)−

R+1

X

w=0

n G^(w,0)(x,1)(−1)^w

w! n^−w/αµ_wM_n,ξ

(105)

with M_n,ξ(x) =H_n,ξ(x)−H_n,0(x).

Lemma 4.5. For the function W_r,n,ξ defined by (102) the following holds true.

(i) W_r,n,ξ is absolutely continuous and differentiable on R with W_r,n,ξ(x) = 0 for x <0.

(ii) There exists such constant W >0 that Wr,n,ξ(x)≤W for all x∈R. (iii) lim

x→+∞W_r,n,ξ(x) = 0.

4.1. Some auxiliary functions and plan of the proof 73

Proof. (i) According to the definition, W_r,n,ξ is a linear combination of the term

G_α,1(·, n)∗M_n,ξ(xn^1/α) and some derivatives of a stable distribution function G_α,1(x, λ). From Lemma 3.14 it follows that G_α,1(x, λ) is infinitely differentiable with respect tox andλ, and its derivatives are absolutely continuous. The function Gα,1(·, n)∗Mn,ξ is absolutely continuous and differentiable as a convolution of the absolutely continuous and infinitely differentiable functionG_α,1 and the function of bounded variation M_n,ξ. These facts together with the property G_α,1(x, λ) = 0 for x <0 and α < 1 (see Section 3.3) and the fact that M_n,ξ(x) = 0 for x <0 give the statement (i) of the lemma.

(ii) Let us consider Gα,1(·, n) ∗Mn,ξ. Using (97) and the fact that Gα,1 is the distribution function of a stable random variable we have

G_α,1(·, n)∗M_n,ξ(xn^1/α)=

Z +∞

−∞

G_α,1xn^1/α−y, n dM_n,ξ(y)

≤

Z +∞

−∞ 1dM_n,ξ(y)=ν₀(M_n,ξ)≤ν₀(H_n,ξ) =ν_0,n,ξ <∞.

Using the last estimate and estimate (59) from Lemma 3.14 we obtain the inequality from (ii) for all x∈R:

W_r,n,ξ(x)≤

p

X

k=2

|c_k,n|

n^k D_0,k+

p

X

k=0 mk

X

`=1

n

`

!|ck,n−`| n^k

m`,k

X

u=`

pu,`,k

X

v=0

D_u,k+v(`/n)^v

v! n^−u/αC_u,`

+n ν_0,n,ξ+n D_R+1,0 1

(R+ 1)!n^−R+1α µ_R+1,n,ξ+

R+1

X

w=0

n D_w,0 n^−w/α w!

µ_w(M_n,ξ)=:W . Note that all pseudomoments occurring inW are finite (see (94) and (97)).

(iii) Let us show that lim

x→+∞

G_α,1(·, n)∗M_n,ξ(xn^1/α) = 0. Using the definition of G_α,1 and Lemma 4.1 (i) we obtain

x→+∞lim

G_α,1(·, n)∗M_n,ξ(xn^1/α) = lim

x→+∞

Z +∞

−∞

G_α,1xn^1/α−y, n dM_n,ξ(y)

=

Z +∞

−∞ lim

x→+∞G_α,1xn^1/α−y, n dM_n,ξ(y) =

Z +∞

−∞ 1dM_n,ξ(y) =µ₀(M_n,ξ) = 0.

The convergence of all other terms to 0 asx → ∞ can be proved in the same way as in Lemma 3.22 (iii). This completes the proof of the lemma.

Now we are ready to give a plan of the proof of our main result (Theorem (3.26)), which we repeat here for the sake of readability. Recall that

F_n(x) = P X₁+· · ·+X_n≤xn^1/α=F^n∗(xn^1/α).

Theorem. If for r >1 we have 0< γ_r^∗ <∞ and ν₀^∗ <1, then for all x∈R and all integers n≥2 the following inequality holds:

F_n(x)−G_α,1(x)−W^f_r,n(x)≤C(1 +|x|)^−rn^−r−αα 1 +n^αr Q_n, where W^fr,n(x) is defined by (87), Qn =ν₀^∗n−1+sup_|t|>

eε|f(t)|+ 2γ_r^∗n^−r/αn−1 with εedefined by (91) and constant C does not depend on x and n.

Plan of the proof.

In order to estimateF_n(x)−G_α,1(x)−W^f_r,n(x) we add and subtract the auxiliary functions F^n∗_n,ξ and W_r,n,ξ with some fixed ξ ∈ [0,∞), and come to the following inequality.

F_n(x)−G_α,1(x)−W^f_r,n(x)

≤F_n(x)−F^n∗_n,ξn^1/αx

+F^n∗_n,ξn^1/αx−Gα,1(x)−Wr,n,ξ(x)+Wr,n,ξ(x)−W^fr,n(x).

We discuss and estimate each of the three summands on the right-hand side sepa-rately in the subsequent subsections. Combining the results we will then obtain the estimation from the theorem.

Im Dokument Approximations and asymptotic expansions for sums of heavy-tailed random variables (Seite 61-74)