CHARACTERISTIC FUNCTIONS OF RANDOM

VARIABLES ATTRACTED TO 1–STABLE LAWS

Jon Aaronson and Manfred Denker

Abstract. The domain of attraction of a 1-stable law on ℝ^{d} is characterised by the expansions of the characteristic functions of its elements.

§0 introduction

Let X_{1}, X_{2}, … be ℝ^{d}-valued, independent, identically distributed random variables. The distributional limits of \frac{S_{n}−A_{n}}{B_{n}} where A_{n} ∈ ℝ^{d}, B_{n} > 0 are constants, and S_{n} = ∑_{k=1}^{n} X_{k}, are given by the well known stable laws. ([Le], [G-K], [I-L]).

A probability distribution function F on ℝ^{d} is called stable if for all a, b > 0 there are c > 0 and v ∈ ℝ^{d} such that

F_{a} ∗ F_{b}(x) = F_{c}(x − v) (x ∈ ℝ^{d})

where F_{s}(x) = F (x/s) (x ∈ ℝ^{d}, s > 0), and strictly stable if this is true with v = 0.

In this case ([Le]) necessarily a^{p} + b^{p} = c^{p} for some 0 < p ≤ 2, and p is called the order of the stable law F.

A distribution G on ℝ^{d} belongs to the domain of attraction of the stable law F if there are constants A_{n} ∈ ℝ^{d} and B_{n} > 0 such that the distributions \frac{S_{n}−A_{n}}{B_{n}} converge weakly to F where S_{n} = X_{1} + ... + X_{n} and X_{1}, X_{2}, ... are i.i.d. with distribution G.

For p ∈ (0, 2] and d ∈ ℕ we let DA(p, d) be the collection of distribution functions in the domain of attraction of some stable law on ℝ^{d} of order p.

In this paper, we obtain expansions of the characteristic functions of distributions on ℝ^{d} which are in the domain of attraction of a stable law.

In §1 we deal with the case d = 1. The first partial results are in [G-Kor]. The expansions are given fully in [I-L] in case p ≠ 1 (see theorem 1 below).

Our main result is theorem 2 (below) giving the expansions in case p = 1.

In §2 we obtain as corollaries expansions in case d ≥ 2. Other results in this case are to be found in [R], [Me], [K-M], [A-G1] and [A-G2]. .

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 60E07, 60E10, 60F05.

This research was supported by a grant from G.I.F., the German-Israel Foundation for Scientific Research and Development. .

A stable law of order p on ℝ has a characteristic function ψ of form

log ψ(t) = itγ − c|t|^{p}[1 − iβsgn(t) tan(\frac{pπ}{2} )] (p ≠ 1),

and

Re log ψ(t) = −c|t|, Im log ψ(t) = t(γ + \frac{2βc}{π} log(1/|t|)) (p = 1)

where c > 0, β, γ ∈ ℝ are constants ([Le]).

The form of the characteristic functions of stable laws on ℝ^{d} was obtained by Feldheim (see [Fe], [Le] and theorem 2.3.1 in [S-T]):

To each stable law of order p on ℝ^{d} there corresponds a finite measure ν on S^{d−1} (called the spectral measure) and μ ∈ ℝ^{d} (called the translate) so that the characteristic function ψ has the form

(1a)  log ψ(u) = i⟨u, μ⟩ − ∫_{S^{d−1}} |⟨u, s⟩|^{p}(1 − isgn(⟨s, u⟩) tan(\frac{pπ}{2} ))ν(ds)

for p ≠ 1 and

(1b)  log ψ(u) = i⟨u, μ⟩ − ∫_{S^{d−1}} |⟨u, s⟩|(1 + i\frac{2}{π} sgn(⟨u, s⟩) log(|⟨u, s⟩|))ν(ds)

for p = 1. Evidently a stable law on ℝ^{d} has a density if and only if the support of its spectral measure is not contained in a proper subspace of ℝ^{d}, and in this case we say that both the stable law, and the spectral measure are nondegenerate.

Clearly, the stability of a ℝ^{d}-valued random variable Z implies that of its inner products ⟨Z, u⟩, (u ∈ ℝ^{d}).

An example of Marcus ([Ma]) shows that the converse of this is false without additional assumptions.

According to theorems 2.1.2 and 2.1.5 in [S-T], the ℝ^{d}-valued random variable Z is strictly stable (stable with order ≥ 1) if its inner products ⟨Z, u⟩, (u ∈ ℝ^{d}) are strictly stable on ℝ (stable on ℝ with order ≥ 1).

The first characterisations of domains of attraction were in terms of the tails of the distributions concerned.

In the unidimensional case for p < 2 ([G-K]): the distribution function G ∈DA(p, 1) iff there is a function L : ℝ_{+} → ℝ_{+}, slowly varying at ∞ (see [F]), and constants c_{1}, c_{2} ≥ 0, c_{1} + c_{2} > 0 such that

L_{1}(x) := x^{p}(1 − G(x)) = (c_{1} + o(1))L(x)
(2)  L_{2}(x) := x^{p}G(−x) = (c_{2} + o(1))L(x) as x → +∞.

The results of [G-K] were generalised to ℝ^{d} in [R] (see also [Me]), to Hilbert space in [K-M], and to Banach space in [A-G1].

The authors would like to thank I.A. Ibragimov for a helpful conversation.

§1 unidimensional characterisation

The characteristic function ψ of G ∈DA(p, 1) is considered in [G-Kor] and [I-L].

In [G-Kor], DA(p, 1) is characterised in terms of ψ(t).

In [I-L], the asymptotic expansion of log ψ(t) around 0 is established with error small when compared to Prob. (|Z| > 1/|t|) = |t|^{p}L(1/|t|) as t → 0. Here, Z is a G-distributed random variable, and G ∈DA(p, 1) (p ≠ 1) satisfies (2) with the slowly varying function L. Specifically:

Theorem 1 ([I-L, theorem 2.6.5]).

Suppose that G satisfies (2) with p ≠ 1, then

log ψ(t) = itγ − c|t|^{p}L(|t|^{−1})[1 − iβsgn(t) tan(\frac{pπ}{2} )]

where

β = \frac{c_{1} − c_{2}}{c_{1} + c_{2}} , c = Γ(1 − p)(c_{1} + c_{2}) cos(\frac{pπ}{2} ), γ = \{ \begin{matrix}0 0 < p < 1 \\ ∫ xG(dx) 1 < p ≤ 2. \end{matrix}

The expansion of the characteristic function when p = 1 is also treated in [I-L] for a limited class of slowly varying functions L, namely those where

∫_{0}^{λ} \frac{xL(x)dx}{1 + x^{2}} = L(λ)(log λ + o(1))

as λ → ∞ (c.f. theorem 2 here, theorem 2.6.5 there, and formula (2.6.34) there). As can be easily checked, the functions L(x) ∼ (log x)^{a} (a ∈ ℝ), and L(x) ∼ e^{(log x)^{a}} (0 < a < 1) are slowly varying functions not in this class.

Theorem 2.

Suppose that G satisfies (2) with p = 1, then

Re log ψ(t) = −c|t|L(|t|^{−1}),

Im log ψ(t) = tγ + \frac{2βc}{π}CtL(1/|t|) + t(H_{1}(1/|t|) − H_{2}(1/|t|)),

H_{j}(λ) = ∫_{0}^{λ} \frac{xL_{j}(x)dx}{1 + x^{2}} + o(L(λ)) as λ → ∞ (j = 1, 2),

C = ∫_{0}^{∞}(cos y − \frac{1}{1 + y^{2}} )\frac{dy}{y} ,

and the constants c > 0, β, γ ∈ ℝ are defined by

β = \frac{c_{1} − c_{2}}{c_{1} + c_{2}} , c = \frac{(c_{1} + c_{2})π}{2} ,

γ = ∫_{−∞}^{∞}(\frac{x}{1 + x^{2}} + sgn(x) ∫_{0}^{|x|} \frac{2u^{2}}{(1 + u^{2})^{2}} du)G(dx)

Remark 1. Note that H_{1}(λ) = ∫_{0}^{λ} \frac{x^{2}P (Z>x)dx}{1+x^{2}}, whence

H_{1}(λ) − H_{2}(λ) = E([|Z| ∧ λ − tan^{−1}(|Z| ∧ λ)]sgn (Z)) = E((|Z| ∧ λ) sgn (Z)) + O(1)

as λ → ∞ where Z is G-distributed and H_{1}, H_{2} are as in theorem 2.

Remark 2.

From this representation of the characteristic function of distributions in DA(p, 1) one deduces existence of a p-stable random variable Y, and constants A_{n}, B_{n} ∈ ℝ, B_{n} > 0 so that \frac{S_{n}−A_{n}}{B_{n}} → Y in distribution. These constants (unique up to o(B_{n}) as n → ∞) are given by

nL(B_{n}) = B_{n}^{p}, A_{n} = \{ \begin{matrix}0 0 < p < 1, \\ γn 1 < p ≤ 2, \\ γn + n(H_{1}(B_{n}) − H_{2}(B_{n})) p = 1. \end{matrix}

To see this in case p = 1 write

log E(e^{it( \frac{Sn−An}{Bn})}) = −\frac{itA_{n}}{B_{n}} + n log ψ(\frac{t}{B_{n}}) := α_{n}(t) + iβ_{n}(t),

then

α_{n}(t) = −c\frac{n|t|}{B_{n}}L(\frac{B_{n}}{|t|}) → −c|t| as n → ∞,

and

β_{n}(t) = \frac{t(H_{1}(B_{n}/|t|) − H_{1}(B_{n}))}{L(B_{n})} − \frac{t(H_{2}(B_{n}/|t|) − H_{2}(B_{n}))}{L(B_{n})} + \frac{2βctCL(B_{n}/|t|)}{πL(B_{n})}.

Now for j = 1, 2 and k > 1 (see (5) in lemma 3 below),

H_{j}(kλ) − H_{j}(λ) = c_{j}L(λ) log k + o(L(λ)) as λ → ∞.

Thus with k = 1/|t|

β_{n}(t) → t(c_{1} − c_{2}) log \frac{1}{|t|} + \frac{2βcCt}{π} = \frac{2βct}{π}(log \frac{1}{|t|} + C) as n → ∞.

Thus, the above representation is a characterization of DA(p, 1).

Remark 3.

We note that the expansion of ψ(t) around 0 up to o(|t|^{p}L(1/|t|)) is determined entirely by the asymptotic equivalence class of the slowly varying function L and the constants c_{1}, c_{2} ≥ 0 for G satisfying (2) with p ≠ 1.

This is not the case when p = 1 as shown by the following examples.

There is a distribution G so that

L_{1}(x) := x(1 − G(x)) = (log x)^{2} + (log x)^{\frac{3}{2}} + O(1)
L_{2}(x) := xG(−x) = (log x)^{2} + O(1) as x → +∞.

Here, L(λ) = (log λ)^{2}, p = c_{1} = c_{2} = 1, and one calculates from theorem 2 that

Im log ψ(t) = \frac{4}{5π} L(\frac{1}{|t|})^{\frac{5}{4}} + o(L(\frac{1}{|t|})) as t → 0.

On the other hand, there is a symmetric distribution satisfying

L_{1}(x) = L_{2}(x) = (log x)^{2} + O(1) as x → +∞

for which also L(λ) = (log λ)^{2}, and p = c_{1} = c_{2} = 1; but here (owing to symmetry)

Im log ψ(t) ≡ 0.

Proof of theorem 2. Assume that G is represented in the form (2).

For x > 0 define distribution functions G_{j} (j = 1, 2) on IR_{+} by

G_{1}(x) = G(x) − G(0), and G_{2}(x) = G(0) − G(−x).

We have that

G_{j}(∞) − G_{j}(x) = \frac{L_{j}(x)}{x} = \frac{(c_{j} + o(1))L(x)}{x}.

Write

∫ (1 − e^{itx} + \frac{itx}{1 + x^{2}} ) G(dx)

= ∫_{0}^{∞}(1 − e^{itx} + \frac{itx}{1 + x^{2}} ) G_{1}(dx) + ∫_{0}^{∞}(1 − \frac{itx}{1 + x^{2}} − e^{−itx}) G_{2}(dx)

and let

γ_{j} = ∫_{0}^{∞} \frac{2x^{2}}{(1 + x^{2})^{2}} (G_{j}(∞) − G_{j}(x))dx = ∫_{0}^{∞} \frac{2xL_{j}(x)dx}{(1 + x^{2})^{2}}.

Integration by parts gives

∫_{0}^{∞}(1 − e^{−(−1)^{j}itx} − (−1)^{j} \frac{itx}{1 + x^{2}} )G_{j}(dx)

= (−1)^{j}it ∫_{0}^{∞}(e^{−(−1)^{j}itx} − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{L_{j}(x)dx}{x}

= |t| ∫_{0}^{∞} sin(|t|x)\frac{L_{j}(x)dx}{x} + (−1)^{j}it ∫_{0}^{∞}(cos(tx) − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{L_{j}(x)dx}{x}.

Changing variables, we obtain that

∫_{0}^{∞} sin(|t|x)\frac{L_{j}(x)dx}{x} = ∫_{0}^{∞} sin(x)\frac{L_{j}(x/|t|)dx}{x},

∫_{0}^{∞}(cos(tx) − \frac{1}{1 + (tx)^{2}} )\frac{L_{j}(x)dx}{x} = ∫_{0}^{∞}(cos(x) − \frac{1}{1 + x^{2}} )\frac{L_{j}(x/|t|)dx}{x}.

By lemma 1 (below) we see that

∫_{0}^{∞} sin(|t|x)\frac{L_{j}(x)dx}{x} = (1 + o(1))L_{j}(\frac{1}{|t|})\frac{π}{2} .

Now

∫_{0}^{∞}(cos(tx) − \frac{1 − x^{2}}{(1 + x^{2})^{2}} )\frac{L_{j}(x)dx}{x} =

∫_{0}^{∞}(cos(tx) − \frac{1}{1 + (tx)^{2}} )\frac{L_{j}(x)dx}{x} + ∫_{0}^{∞} \frac{x(1 − t^{2})L_{j}(x)dx}{(1 + x^{2})(1 + (tx)^{2})} + ∫_{0}^{∞} \frac{2xL_{j}(x)dx}{(1 + x^{2})^{2}}

= ∫_{0}^{∞}(cos(tx) − \frac{1}{1 + (tx)^{2}} )\frac{L_{j}(x)dx}{x} + ∫_{0}^{∞} \frac{x(1 − t^{2})L_{j}(x)dx}{(1 + x^{2})(1 + (tx)^{2})} + γ_{j}.

By lemma 2 (below)

∫_{0}^{∞}(cos(tx) − \frac{1}{1 + (tx)^{2}} )\frac{L_{j}(x)dx}{x} = CL_{j}(\frac{1}{|t|}) + o(L(\frac{1}{|t|})).

Set

H̃_{j}(λ) := ∫_{0}^{∞} \frac{xL_{j}(x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )}.

By lemma 3 (below), H̃_{j}(λ) = H_{j}(λ) + o(L(λ)) as λ → ∞. Putting everything together we obtain

∫_{0}^{∞}(1 + \frac{itx}{1 + x^{2}} − e^{itx})G_{1}(dx) + ∫_{0}^{∞}(1 − \frac{itx}{1 + x^{2}} − e^{−itx})G_{2}(dx)

= L(\frac{1}{|t|})|t|(c_{1} + c_{2})π/2 − itL(1/|t|)(c_{1} − c_{2})C

− it(H̃_{1}(1/|t|) − H̃_{2}(1/|t|)) − it(γ_{1} − γ_{2}) + o(|t|L(\frac{1}{|t|}))

= L(\frac{1}{|t|})|t|(c_{1} + c_{2})π/2 − itL(1/|t|)(c_{1} − c_{2})C

− it(H_{1}(1/|t|) − H_{2}(1/|t|)) − it(γ_{1} − γ_{2}) + o(|t|L(\frac{1}{|t|}))

and hence theorem 2. □

We conclude this section by collecting the lemmas on slowly varying functions needed for theorem 2.

Assume that h : ℝ_{+} → ℝ_{+} is locally integrable, slowly varying at infinity, and such that u ↦ \frac{h(u)}{u} is a non-increasing function.

C.F. OF RANDOM VARIABLES ATTRACTED TO 1–STABLE LAWS 7

Lemma 1.

∫_{0}^{∞} \frac{sin y}{y} h(\frac{y}{t})dy = (1 + o(1))h(\frac{1}{t})\frac{π}{2} .

Proof. As the proof of lemma 2.6.1 in [I-L]. □ Lemma 2.

∫_{0}^{∞} [cos y − \frac{1}{1 + y^{2}} ] \frac{1}{y} h(\frac{y}{t})dy = (1 + o(1))h(\frac{1}{t}) ∫_{0}^{∞} [cos y − \frac{1}{1 + y^{2}} ] \frac{1}{y} dy.

Proof. We first split the region of integration into four parts: I_{1} = [Δ_{1}, ∞), I_{2} = [δ, Δ_{1}), I_{3} = [tΔ_{2}, δ) and I_{4} = [0, tΔ_{2}) where δ < 1 < Δ_{1} = (N − \frac{1}{2})π (N ∈ ℕ).

Since | ∫_{[Δ_{1}+nπ,Δ_{1}+(n+1)π]} cos y \frac{h(y/t)dy}{y} | decreases in n,

|∫_{I_{1}} cos y \frac{h(y/t)dy}{y} | ≤ \frac{πh(Δ_{1}/t)}{Δ_{1}} ∼ \frac{πh(1/t)}{Δ_{1}}.

Also,

∫_{I_{1}} \frac{1}{1 + y^{2}} \frac{h(y/t)dy}{y} ≤ \frac{h(Δ_{1}/t)}{Δ_{1}}π ∼ \frac{πh(1/t)}{Δ_{1}}.

Since for x ∈ [Δ_{2}t, δ)

\frac{h(x/t)}{h(1/t)} = (1 + o(1)) exp[∫_{x/t}^{1/t} \frac{ϵ(s)}{s}ds] = exp[o(− log x)] ≤ x^{−1/2}

for t small enough,

|∫_{I_{3}}(1 − cos y)h(y/t)\frac{dy}{y} | ≤ h(1/t) ∫_{0}^{δ} \frac{1 − cos y}{y^{3/2}} dy ≤ 2h(1/t) ∫_{0}^{δ} y^{1/2}dy.

For the function h is bounded on I_{4}, hence for t small enough

|∫_{I_{4}}(1 − cos y)h(y/t)\frac{dy}{y} | ≤ sup_{I_{4}} h ∫_{0}^{Δ_{2}t}(1 − cos y)\frac{dy}{y}

= O(t^{2}) = o(h_{1}(1/t))

For δ ≤ x ≤ Δ_{1} we have (uniformly in x) by the slow variation property of h

lim_{t→0} \frac{h(x/t)}{h(1/t)} = 1.

8 JON AARONSON AND MANFRED DENKER

It follows that

|∫_{I_{2}}(1 − cos y) [h(y/t) − h(1/t)] \frac{dy}{y} |

≤ h(1/t) [ sup_{δ≤x≤Δ_{1}} |\frac{h(x/t)}{h(1/t)} − 1|] ∫_{δ}^{Δ_{1}} \frac{dy}{y}

= o(h(1/t)).

Applying the estimates for I_{1}, I_{3}, and I_{4} with h = 1 it follows that

∫_{0}^{Δ_{1}}(1 − cos y)\frac{h(y/t)}{y}dy − h(1/t) ∫_{0}^{Δ_{1}}(1 − cos y)\frac{dy}{y} = o(h(1/t))

and

| ∫_{Δ_{1}}^{∞} \frac{1}{y} [cos y − \frac{1}{1 + y^{2}} ] [h(y/t)) − h(1/t)] dy| <_{∼} \frac{2πh(1/t)}{Δ_{1}}.

Since

∫_{0}^{1} |\frac{h(uz)}{h(z)} − 1|du → 0 as z → ∞,

∫_{0}^{Δ_{1}} \frac{y}{1 + y^{2}} [h(y/t) − h(1/t)] dy = ∫_{0}^{1} \frac{Δ_{1}^{2}z}{1 + (Δ_{1}z)^{2}} [h(Δ_{1}z/t) − h(1/t)] dz

≤ h(1/t) ∫_{0}^{1} |\frac{h(Δ_{1}z/t)}{h(1/t)} − 1| dz = o(h(1/t))

as t → 0.

The lemma follows from

∫_{0}^{∞} [cos y − \frac{1}{1 + y^{2}} ] (h(y/t) − h(1/t)) dy

= ∫_{Δ_{1}}^{∞} [cos y − \frac{1}{1 + y^{2}} ] (h(y/t) − h(1/t)) dy

− ∫_{0}^{Δ_{1}}(1 − cos y)\frac{h(y/t)}{y}dy − h(1/t) ∫_{0}^{Δ_{1}}(1 − cos y)\frac{dy}{y}

+ ∫_{0}^{Δ_{1}} \frac{y}{1 + y^{2}} [h(y/t) − h(1/t)] dy

<_{∼} \frac{2πh(1/t)}{Δ_{1}}

as Δ_{1} → ∞. □

Lemma 3. Let

H(λ) := ∫_{0}^{λ} \frac{xh(x)dx}{1 + x^{2}};

then H is slowly varying at infinity,

(3)  \frac{h(λ)}{H(λ)} → 0 as λ → ∞,

(4)  H̃(λ) := ∫_{0}^{∞} \frac{xh(x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )} = H(λ) + o(h(λ)) as λ → ∞,

and

(5)  H(kλ) − H(λ) ∼ h(λ) · log k  as λ → ∞.

Remark. Slow variation of H, (3), and (5) are established in lemma 1 of [Par].

Proof.

We first show (5):

H(kλ) − H(λ) = ∫_{λ}^{kλ} \frac{xh(x)dx}{1 + x^{2}} ∼ ∫_{λ}^{kλ} \frac{h(x)dx}{x}

= ∫_{1}^{k} \frac{h(λx)dx}{x} ∼ log k h(λ).

Next, we see that (3) follows from (5) as ∀ M > 1,

\frac{H(λ)}{h(λ)} = \frac{H(e^{M}e^{−M}λ)}{h(λ)} ≥ \frac{H(e^{M}e^{−M}λ) − H(e^{−M}λ)}{h(λ)} ∼ \frac{h(e^{−M}λ)M}{h(λ)} → M

as λ → ∞. It follows from (3) and (5) that H is slowly varying at ∞. To continue, we claim that

(6)  H̃(λ) = ∫_{0}^{λ} \frac{xh(x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )} + \frac{log 2}{2}h(λ) + o(h(λ))  as λ → ∞.

To see this, note that

∫_{λ}^{∞} \frac{xh(x)dx}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )} = ∫_{1}^{∞} \frac{xh(λx)dx}{(\frac{1}{λ^{2}} + x^{2})(1 + x^{2})}

= h(λ) ∫_{1}^{∞} \frac{xdx}{(\frac{1}{λ^{2}} + x^{2})(1 + x^{2})} + h(λ) ∫_{1}^{∞}(\frac{h(λx)}{h(λ)} − 1)\frac{xdx}{(\frac{1}{λ^{2}} + x^{2})(1 + x^{2})}

= \frac{log 2}{2}h(λ) + o(h(λ))

as λ → ∞ by the dominated convergence theorem since |\frac{h(λx)}{h(λ)} − 1| → 0 as λ → ∞ ∀ x > 1

and |\frac{h(λx)}{h(λ)} − 1| ≤ x ∀ x large enough. This establishes (6).

To finish the proof of (4), we note that

\frac{xh(x)}{(1 + x^{2})(1 + \frac{x^{2}}{λ^{2}} )} = \frac{λ^{2}}{λ^{2} − 1}(\frac{xh(x)}{x^{2} + 1} − \frac{xh(x)}{x^{2} + λ^{2}} ),

whence in view of (6),

H̃(λ) = \frac{λ^{2}}{λ^{2} − 1} ∫_{0}^{λ} \frac{xh(x)dx}{x^{2} + 1} − \frac{λ^{2}}{λ^{2} − 1} ∫_{0}^{λ} \frac{xh(x)dx}{x^{2} + λ^{2}} + \frac{log 2}{2}h(λ) + o(h(λ))

Now

\frac{λ^{2}}{λ^{2} − 1} ∫_{0}^{λ} \frac{xh(x)dx}{x^{2} + 1} = H(λ) + O(\frac{H(λ)}{λ^{2}}) = H(λ) + o(h(λ)) as λ → ∞

because both h and H are slowly varying at ∞; and

\frac{λ^{2}}{λ^{2} − 1} ∫_{0}^{λ} \frac{xh(x)dx}{x^{2} + λ^{2}} ∼ ∫_{0}^{λ} \frac{xh(x)dx}{x^{2} + λ^{2}} = ∫_{0}^{1} \frac{xh(λx)dx}{x^{2} + 1} ∼ \frac{log 2}{2}h(λ) as λ → ∞.

Thus

H̃(λ) = H(λ) + o(h(λ)) as λ → ∞

which is (4). □

§2 multidimensional corollaries

Corollary 1. Let 0 < p < 2, p ≠ 1 and G be a distribution function on ℝ^{d}. The following are equivalent:

(A) G belongs to the domain of attraction of the nondegenerate stable law of order p, spectral measure ν and translate μ.

(B) The characteristic function ψ of G has the form

log ψ(tu) = \{ \begin{matrix}−t^{p}L(\frac{1}{t})Φ(u) + it⟨u, μ⟩ + o(t^{p}L(\frac{1}{t})) & if p > 1 \\ −t^{p}L(\frac{1}{t})Φ(u) + o(t^{p}L(\frac{1}{t})) & if p < 1 \end{matrix}

as t → 0^{+}, ∀ u ∈ S^{d−1}, where μ ∈ ℝ^{d}, L is slowly varying at infinity, ν is a nondegenerate finite measure on S^{d−1} and

Φ(u) := ∫_{S^{d−1}} |⟨u, s⟩|^{p}(1 − isgn⟨s, u⟩ tan(\frac{pπ}{2} ))ν(ds).

Proof of corollary 1. (A)⇒(B).

Let X_{1}, X_{2}, ... be i.i.d. with distribution G and A_{n} ∈ ℝ^{d}, B_{n} > 0 such that \frac{S_{n}−A_{n}}{B_{n}} → Z weakly where Z is p-stable. Let u ∈ ℝ^{d}. It follows from Feldheim’s theorem that ⟨u, Z⟩ has a 1-dimensional p-stable distribution with parameters γ′_{u} = ⟨u, μ⟩, c′_{u} = ∫_{S^{d−1}} |⟨u, s⟩|^{p}ν(ds) and

β′_{u} = \frac{1}{c′_{u}} ∫_{S^{d−1}} |⟨u, s⟩|^{p}sgn(⟨u, s⟩)ν(ds).

The characteristic function ψ(tu) of ⟨u, X_{1}⟩ has a form

log ψ(tu) = itγ_{u} − |t|^{p}L_{u}(1/|t|)(1 − iβ_{u}sgn(t) tan(\frac{πp}{2}))

as in theorem 1 with some slowly varying function L_{u} and parameters γ_{u} and β_{u} (we normalize L_{u} so that c_{u} = 1). Hence

it(\frac{nγ_{u}}{B_{n}} − \frac{⟨u, A_{n}⟩}{B_{n}}) − |t|^{p} \frac{n}{B_{n}^{p}} L_{u}(\frac{B_{n}}{|t|} )(1 − iβ_{u}sgn(t) tan(\frac{pπ}{2} ))

→ itγ′_{u} − c′_{u}|t|^{p}(1 − iβ′_{u}sgn(t) tan(\frac{pπ}{2} )).

The parameter γ_{u} must be linear in u if p > 1, since \frac{nγ_{u}−⟨u,A_{n}⟩}{B_{n}} → ⟨u, μ⟩ and \frac{n}{B_{n}} → ∞. In case p < 1, γ_{u} can be arbitrary since \frac{n}{B_{n}} → 0. Moreover, \frac{n}{B_{n}^{p}} L_{u}(B_{n}) converges to c′_{u} and β_{u} = β′_{u}. Setting L(t) = \frac{1}{c′_{u}} L_{u}(t) for some fixed u we obtain for v ∈ IR^{d}

lim_{n→∞} \frac{L(B_{n})}{L_{v}(B_{n})} = lim_{n→∞} \frac{(n/B_{n}^{p})L_{u}(B_{n})}{c′_{u}(n/B_{n}^{p})L_{v}(B_{n})} = 1/c′_{v},

hence L_{v}(λ) ∼ c′_{v}L(λ) as λ → ∞.

(B)⇒(A).

Conversely, if the characteristic function ψ of G is as in (B), then for every u ∈ ℝ^{d} the characteristic functions of Y_{n}^{(u)} = B_{n}^{−1} ∑_{k=1}^{n}(⟨u, X_{k}⟩−⟨A_{n}, u⟩) converges, where X_{1}, X_{2}, ... are i.i.d. with distribution G, where B_{n} is defined by nL(B_{n}) = B_{n}^{p} and where A_{n} = 0 if p < 1 and A_{n} = nμ if p > 1.

It follows that the characteristic functions of \frac{S_{n}−A_{n}}{B_{n}} converge (necessarily to a characteristic function), such that the limit variable Z has all distributions ⟨u, Z⟩ (u ∈ IR^{d}) p-stable. Thus Z is stable itself if p > 1. In case p < 1 we note that Z has a characteristic function of the form (1a) with μ = 0 and is strictly stable. □

If G is a distribution function on ℝ^{d} we define G_{u}(·) to be the distribution function of ⟨u, Z⟩ where Z is a random variable with distribution G.

Corollary 2. (A) If a distribution function G on ℝ^{d} belongs to the domain of attraction of the nondegenerate stable law of order 1, spectral measure ν and translate μ, then its characteristic function ψ has the form

Re log ψ(tu) = −tL(\frac{1}{t}) ∫_{S^{d−1}} |⟨u, s⟩|ν(ds) + o(tL(\frac{1}{t})),

Im log ψ(tu) = tH_{u}(\frac{1}{t}) + tL(\frac{1}{t})\frac{2C}{π} ∫_{S^{d−1}}⟨u, s⟩ν(ds) + tγ_{u} + o(tL(\frac{1}{t})) (7)

as t → 0^{+} ∀ u ∈ S^{d−1}, where : L is slowly varying at infinity, C = ∫_{0}^{∞}(cos y − \frac{1}{1+y^{2}} )\frac{dy}{y}, and where

H_{u}(x) = ∫_{0}^{x} \frac{v(1 − G_{u}(v) − G_{u}(−v))}{1 + v^{2}}dv

has a representation

(8)  H_{u}(λ) = ⟨u, Γ_{λ}⟩ − \frac{2L(λ)}{π} ∫_{S^{d−1}}⟨u, s⟩ log(|⟨u, s⟩|)ν(ds) − γ_{u} + o(L(λ))

for some Γ_{λ} ∈ ℝ^{d} and satisfies

(9)  H_{u}(kλ) − H_{u}(λ) ∼ \frac{2}{π} L(λ) ∫_{S^{d−1}}⟨u, s⟩ν(ds) log k

as λ → ∞.

(B) Let the characteristic function ψ of a distribution G on ℝ^{d} satisfy (7) for some γ_{u} ∈ ℝ, some finite measure ν on S^{d−1}, some slowly varying function L and some functions H_{u} with representation (8) and satisfying (9). Then G belongs to the domain of attraction of a nondegenerate stable law of order 1.

Proof of corollary 2. (A) As before, let X_{1}, X_{2}, ... be i.i.d. with distribution G and A_{n} ∈ ℝ^{d}, B_{n} > 0 such that \frac{S_{n}−A_{n}}{B_{n}} → Z weakly where Z is 1-stable. Let u ∈ ℝ^{d}. It follows from Feldheim’s theorem that ⟨u, Z⟩ has a 1-dimensional 1-stable distribution with parameters

γ′_{u} = ⟨u, μ⟩ − \frac{2}{π} ∫_{S^{d−1}}⟨u, s⟩ log(|⟨u, s⟩|)ν(ds)

c′_{u} = ∫_{S^{d−1}} |⟨u, s⟩|ν(ds), β′_{u} = \frac{1}{c′_{u}} ∫_{S^{d−1}}⟨u, s⟩ν(ds).

By theorem 2, the characteristic function ψ(tu) of ⟨u, X_{1}⟩ has a form

log ψ(tu) = −|t|L_{u}(\frac{1}{|t|})+

itγ_{u} + it\frac{2β_{u}C}{π} L_{u}(\frac{1}{|t|}) + it(H_{1u}(1/|t|) − H_{2u}(1/|t|)) + o(|t|L_{u}(1/|t|))

where

H_{ju}(λ) = ∫_{0}^{λ} \frac{xL_{ju}(x)}{1 + x^{2}}dx

L_{ju}(x) = \{ \begin{matrix}x(1 − G_{u}(x)) & if j = 1 \\ xG_{u}(−x) & if j = 2 \end{matrix}

for some parameters γ_{u}, β_{u} and slowly varying functions L_{u} (normalised so that c_{u} = 1), L_{ju}. Also note that by theorem 2 L_{ju}(x) = (c_{ju} + o(1))L_{u}(x) with c_{1u} + c_{2u} = 2/π. Set H_{u} = H_{1u} − H_{2u}.

From the assumed convergence of characteristic functions, we have that

Re n log ψ(\frac{tu}{B_{n}}) ∼ \frac{nL_{u}(B_{n})|t|}{B_{n}} → c′_{u}|t|.

As in the proof of corollary 1, there exists a function L so that c′_{v}L ∼ L_{v} for all v ∈ ℝ^{d}. Moreover, using (5) ∀ t ∈ ℝ, as n → ∞,

Im n log ψ(\frac{tu}{B_{n}}) − ⟨A_{n}, u⟩\frac{t}{B_{n}} = \frac{nL_{u}(B_{n})}{B_{n}}(c_{1u} − c_{2u})t log \frac{1}{|t|}+

t(\frac{nγ_{u}}{B_{n}} − \frac{⟨A_{n}, u⟩}{B_{n}} + \frac{nH_{u}(B_{n})}{B_{n}} + \frac{2Cnβ_{u}L_{u}(B_{n})}{πB_{n}}) + o(1)

→ tγ′_{u} + \frac{2β′_{u}c′_{u}t}{π} log \frac{1}{|t|}.

Equating coefficients of t, and t log \frac{1}{|t|}, we see that

\frac{nL_{u}(B_{n})}{B_{n}}(c_{1u} − c_{2u}) → \frac{2β′_{u}c′_{u}}{π}

and

\frac{n}{B_{n}} (H_{u}(B_{n}) + \frac{2Cβ_{u}}{π} L_{u}(B_{n}) + γ_{u} − ⟨u, A_{n}/n⟩) → γ′_{u}

as n → ∞.

Hence c′_{u}(c_{1u} − c_{2u}) = c′_{u}β_{u}2/π = c′_{u}2β′_{u}/π and β_{u} = β′_{u}.

To conclude, we determine the conditions for H_{u} and γ_{u}. Since c′_{u}L ∼ L_{u} and since L_{u} is slowly varying,

H_{u}(B_{n}) + \frac{2Cβ′_{u}c′_{u}}{π} L(B_{n}) + γ_{u} − ⟨u, A_{n}/n⟩

− ⟨u, \frac{B_{n}μ}{n}⟩ + \frac{2B_{n}}{nπ} ∫_{S^{d−1}}⟨u, s⟩ log(|⟨u, s⟩|)ν(ds)

= o(\frac{B_{n}}{n}),

or (because β′_{u}c′_{u} is linear in u and nL(B_{n}) ∼ B_{n})

H_{u}(B_{n}) = ⟨u, Γ_{B_{n}}⟩ − \frac{2L(B_{n})}{π} ∫_{S^{d−1}}⟨u, s⟩ log(|⟨u, s⟩|)ν(ds) − γ_{u} + o(L(B_{n})),

where

Γ_{B_{n}} = \frac{A_{n}}{n} + μL(B_{n}) − \frac{2CL(B_{n})}{π} ∫_{S^{d−1}}⟨·, s⟩ν(ds).

We obtain the expansion for H_{u}(λ) (B_{n} ≤ λ < B_{n+1}) from

H_{u}(λ) − H_{u}(B_{n}) = H_{1u}(λ) − H_{1u}(B_{n}) − [H_{2u}(λ) − H_{2u}(B_{n})]

∼ log(\frac{λ}{B_{n}} )(L_{1u}(λ) − L_{2u}(λ)) + o(L(λ)) = o(L(λ))

and

H_{u}(λ) = H_{u}(B_{n}) + H_{u}(λ) − H_{u}(B_{n}) = H_{u}(B_{n}) + o(L(λ))

= ⟨u, Γ_{B_{n}}⟩ − \frac{2L(λ)}{π} ∫_{S^{d−1}}⟨u, s⟩ log(|⟨u, s⟩|)ν(ds) − γ_{u} + o(L(λ)),

since

1 ≤ \frac{λ}{B_{n}} ≤ \frac{B_{n+1}}{B_{n}} ∼ \frac{(n + 1)L(B_{n+1})}{nL(B_{n})} → 1.

(8) follows setting Γ_{λ} = Γ_{B_{n}} if B_{n} ≤ λ < B_{n+1}. Finally, (9) holds because

H_{u}(kλ) − H_{u}(λ) ∼ log(k)(L_{1u}(λ) − L_{2u}(λ)) ∼ log(k)(c_{1u} − c_{2u})L_{u}(λ)

∼ log(k)(c_{1u} − c_{2u})c′_{u}L(λ) = \frac{2}{π} c′_{u}β′_{u} log(k)L(λ).

(B) Conversely, if the characteristic function ψ of G is as in (B), then for every u ∈ ℝ^{d} the characteristic functions of

Y_{n}^{(u)} = B_{n}^{−1} ∑_{k=1}^{n}(⟨u, X_{k}⟩ − ⟨A_{n}, u⟩)

converges, where X_{1}, X_{2}, ... are i.i.d. with distribution G, where B_{n} is defined by nL(B_{n}) = B_{n} and where

A_{n} = nΓ_{B_{n}} + \frac{2CnL(B_{n})}{π} ∫_{S^{d−1}}⟨·, s⟩ν(ds).

Let c′_{u} = ∫_{S^{d−1}} |⟨u, s⟩|ν(ds) be defined as before. We have that

log(ψ(\frac{tu}{B_{n}} )^{n}e^{− \frac{it⟨u,An⟩}{Bn}} ) → −|t|c′_{u} − it\frac{2}{π} ∫_{S^{d−1}}⟨u, s⟩ log |⟨tu, s⟩|ν(ds).

□

Example. Let 0 < p < 2, ν ∈ P(S^{d−1}) be nondegenerate, and let L be slowly varying at ∞.

If Y ∈ DA (p, 1), Y > 0 with tails given by P (Y > λ) = \frac{2L(λ)}{πλ^{p}}, and Z is a ν-distributed random variable on S^{d−1} independent of Y, then X := Y Z is in the domain of attraction of a nondegenerate stable law of order p on ℝ^{d}, and with spectral measure ν.

This follows from (and illustrates) corollaries 1 and 2. Indeed, using the notation

ψ_{U}(u) := − log(E(e^{i⟨U,u⟩}), we have that for u ∈ S^{d−1} and t > 0

ψ_{X}(tu) = E(ψ_{Y} (⟨Z, tu⟩) + O(ψ_{Y} (⟨Z, tu⟩)^{2})) = E(ψ_{Y} (⟨Z, tu⟩)) + o(t^{p}L(1/t)),

as t → 0, whence by [I-L] for p ≠ 1

ψ_{X}(tu) =

itγ⟨u, E(Z)⟩ − t^{p}L(1/t) ∫_{S^{d−1}} |⟨u, s⟩|^{p}(1 − isgn(⟨s, u⟩) tan(\frac{pπ}{2} ))ν(ds) + o(t^{p}L(1/t))

as t → 0; and by theorem 2 for p = 1

Re ψ_{X}(tu) = −tL(1/t) ∫_{S^{d−1}} |⟨s, u⟩|dν(s) + o(tL(1/t)),

Im ψ_{X}(tu) = tγ⟨u, E(Z)⟩ + t(H(1/t) + \frac{2C}{π} L(1/t)) ∫_{S^{d−1}}⟨s, u⟩dν(s)

+ tL(1/t)\frac{2}{π} ∫_{S^{d−1}}⟨s, u⟩ log \frac{1}{|⟨s, u⟩|}dν(s) + o(tL(1/t))

as t → 0, where H(λ) := ∫_{0}^{λ} \frac{2xL(x)dx}{π(1+x^{2})} and where γ := E(\frac{Y}{1+Y^{2}} + ∫_{0}^{Y} \frac{2u^{2}}{(1+u^{2})^{2}} du).

If, in the example Y was not chosen positive, but satisfying (2) with constants c, c_{1}, c_{2}, then the spectral measure of X is given by

ν^{∗}(A) = c_{1}ν(A) + c_{2}ν(−A) (A ∈ B(S^{d−1}).

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Aaronson: School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel. E-mail: aaro@math.tau.ac.il

Denker: Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, 37083 Göttingen, Germany E-mail: denker@namu01.gwdg.de