Convergence rates in precise asymptotics II

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U.U.D.M. Report 2011:15

Department of Mathematics Uppsala University

Convergence rates in precise asymptotics II

Allan Gut and Josef Steinebach

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Convergence rates in precise asymptotics II

Allan Gut Uppsala University

Josef Steinebach University of Cologne

Abstract

Let X1, X2, . . . be independent, identically distributed (i.i.d.) random variables with partial sums Sn, n ≥ 1. The now classical Baum-Katz problem concerns finding necessary and sufficient moment conditions for the convergence of P∞

n=1n^r/p−2P (|Sn| ≥ εn^1/p) for fixed ε > 0. A now equally classical paper by Heyde in 1975 initiated what has later been called precise asymptotics, namely asymptotics for the same sum (for the case r = 2 and p = 1) when, instead, ε & 0. In a predecessor of this paper we extend a result due to Klesov (1994), in which he determined the convergence rate in Heyde’s theorem, to the case r ≥ 2, 0 < p < 2. The present paper is devoted to the case when the summands belong to the normal domain of attraction of a stable distribution with index α ∈ (1, 2], in particular to the analog related to Spitzer’s 1956-theorem.

1 Introduction

The point of departure of this note is the following part of the main result in Baum and Katz [1].

Theorem 1.1 Let r > 0, 0 < p < 2 and r ≥ p. Suppose that X, X₁, X₂, . . . are i.i.d. random variables with E |X|^r< ∞ and, if r ≥ 1, E X = 0, and set S_n=Pn

k=1X_k, n ≥ 1. Then

∞

X

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) < ∞ for all ε > 0. (1.1) Conversely, if the sum is finite for some ε > 0, then E|X|^r< ∞ and, if r ≥ 1, E X = 0. In particular, the conclusion then holds for all ε > 0.

One problem of interest is to examine the rate at which the above probabilities tend to one as ε & 0, more precisely, to find some normalizing function of ε for which the sum in Theorem 1.1, premultiplied by this very function, has a nondegenerate limit as ε & 0. Toward that end, Heyde [8] proved that

lim

ε&0ε²

∞

X

n=1

P (|Sn| ≥ εn) = EX², (1.2)

whenever EX = 0 and EX² < ∞. Remaining values of r and p have later been taken care of in [2, 10, 3], and have been coined under the heading “Precise asymptotics for ...”.

The following theorem, due du Klesov [9], provides information about the rate of convergence in Heyde’s result (1.2).

Theorem 1.2 Let X, X1, X2, . . . be i.i.d. random variables, and set Sn=Pn

k=1Xk, n ≥ 1.

(a) If X is normal with mean 0 and variance σ²> 0, then

ε&0lim

X^∞

n=1

P (|Sn| ≥ εn) −σ² ε²

= −1 2. (b) If E X = 0, E X²= σ²> 0, and E|X|³< ∞, then

lim

ε&0ε^3/2X^∞

n=1

P (|S_n| ≥ εn) −σ² ε²

= 0.

AMS 2000 subject classifications. Primary 60F15, 60G50; Secondary 60F05.

Keywords and phrases. Law of large numbers, Baum-Katz, precise asymptotics, convergence rates.

Abbreviated title. Convergence rates in precise asymptotics.

Date. August 18, 2011

1

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2 Allan Gut and Josef Steinebach

In [4] we extended Klesov’s theorem to the case r ≥ 2, 0 < p < 2 as follows.

Theorem 1.3 Let r ≥ 2 and 0 < p < 2. Suppose that X, X1, X2, . . . are i.i.d. random variables, and set Sn =Pn

k=1Xk, n ≥ 1. Let Y be normal with mean 0 and variance σ²> 0.

(a) If E X = 0, E X² = σ² > 0, and E|X|^q < ∞ for some r < q ≤ 3, such that q(r − p)/(q − p) >

2(r − 2p)/(2 − p), if r > 2p, then

lim

ε&0εq(r−p)/(q−p)X^∞

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p· ε−2(r−p)/(2−p)E|Y |2(r−p)/(2−p)

= 0.

(b) If E X = 0, E X² = σ² > 0, and E|X|^q < ∞ for some q ≥ 3, q > (2r − 3p)/(2 − p), such that 2q(r − p)/(p + q(2 − p)) > 2(r − 2p)/(2 − p), if r > 2p, then

lim

ε&0ε2q(r−p)/(p+q(2−p))X^∞

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p · ε−2(r−p)/(2−p)E|Y |2(r−p)/(2−p)

= 0.

The first step in the proof of this result amounts to proving the following proposition, which extends part (a) in [9]. The theorem itself is the analog of his part (b).

Proposition 1.1 Let 0 < p < 2 and r ≥ 2, and suppose that Y , X1, X2, . . . are i.i.d. normal random variables with mean 0 and variance σ²> 0, and set Sn=Pn

k=1Xk, n ≥ 1.

(i) If 0 < r < 2p, then

ε&0lim

X^∞

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) − p

r − p· ε⁻^2(r−p)^2−p E|Y |^2(r−p)^2−p

= 0 .

If, in addition, 2r − 5p + 2 > 0, then

∞

X

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p· ε⁻^2(r−p)^2−p E|Y |^2(r−p)^2−p = O ε⁻^2(r−3p)^2−p . If, in addition, 2r − 7p + 2 > 0, then

lim

ε&0ε^2(r−3p)^2−p X^∞

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) − p

r − p· ε⁻^2(r−p)^2−p E|Y |^2(r−p)^2−p

= −r − 2p

4p · E|Y |^2(r−3p)^2−p ; (ii) If r = 2p, then

lim

ε&0

X^∞

n=1

P (|Sn| ≥ εn^1/p) −1

2 · ε⁻^2−p^2p E|Y |^2−p^2p

= −1 2; (iii) If r > 2p, then

∞

X

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p· ε⁻^2(r−p)^2−p E|Y |^2(r−p)^2−p = O ε⁻^2(r−3p)^2−p . If, in addition, 2r − 7p + 2 > 0, then

lim

ε&0ε^2(r−3p)^2−p X^∞

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) − p

r − p· ε⁻^2(r−p)^2−p E|Y |^2(r−p)^2−p

= −r − 2p

4p · E|Y |^2(r−3p)^2−p . The rate results so far all assume finite variance. It thus remains to investigate the case when the variance is not necessarily finite. Here we shall consider Klesov type rate results for the case when the summands belong to the normal domain of attraction of a stable distribution with index α ∈ (1, 2].

This includes, in particular, in Theorem 2.1 below, results related to Spitzer’s theorem [11], in which he treats case r = p = 1, here however, for the more general case r = 1, 0 < p < 2.

For ease of reference here is a plain convergence result (cf. [3, 10]) that is relevant for our first extension, which concerns the Spitzer case.

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Precise asymptotics; convergence rates 3

Theorem 1.4 Suppose that X, X1, X2, . . . are i.i.d. random variables with mean 0 that belong to the normal domain of attraction of a nondegenerate stable law G with index α ∈ (1, 2], and set Sn = Pn

k=1Xk, n ≥ 1. Then, for 1 ≤ p < α ≤ 2,

ε&0lim 1

− log ε

∞

X

n=1

1

nP (|Sn| ≥ εn^1/p) = αp

α − p. (1.3)

In particular, if Var X = σ²< ∞, the limit exists and equals 2p/(2 − p).

Our second extension will be related to the following result (cf. [3]).

Theorem 1.5 Suppose that X, X₁, X₂, . . . are i.i.d. random variables with mean 0 that belong to the normal domain of attraction of a nondegenerate stable distribution (function) G with index α ∈ (1, 2], and set Sn =Pn

k=1Xk, n ≥ 1. Then, for 1 ≤ p < r < α, lim

ε&0εα(r−p)/(α−p)

∞

X

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) = p

r − p· E|Y |α(r−p)/(α−p), (1.4) where Y is a random variable with distribution G.

2 Results

The first aim of the present paper is to prove a convergence rate result with respect to Theorem 1.4.

For p = 1 this corresponds to the Spitzer case; [11].

Theorem 2.1 Let 1 ≤ p < α ≤ 2, suppose that X, X₁, X₂, . . . are i.i.d. with mean 0 and distribution function F that belong to the normal domain of attraction of a nondegenerate stable distribution (function) G with index α ∈ (1, 2], and set S_n=Pn

k=1X_k, n ≥ 1.

(a) If α = 2, Var (X) = σ², and EX²log(1 + |X|) < ∞, then

lim

ε&0

X^∞

n=1

1

nP (|S_n| ≥ εn^1/p) + 2p

2 − plog ε

= 2(1 − p)

2 − p γ + p

2 − plogσ² 2

− %, (2.1)

where γ is Euler’s constant (= 0.57721 . . . ) and % =P∞

n=1n⁻¹P (S_n= 0).

(b) If 1 < α < 2 andR∞

−∞|x|^α−1|F (x) − G(x)| dx < ∞, then lim

ε&0

X^∞

n=1

1

nP (|S_n| ≥ εn^1/p) + αp

α − plog ε

= αp

α − p· E log |Y | + γ − %, (2.2) where γ and % are as defined above and Y is Stable(α)-distributed with mean 0.

Remark 2.1 (i) It is well-known that % as given in Theorem 2.1 is finite, since the first moment exists (and EX = 0); cf. Spitzer [12], Corollary 3.3.

(ii) Note also that E log |Y | < ∞ in (2.2), since E|Y |^β < ∞ for all 0 < β < α, the density of Y is bounded on [0, 1], andR1

0 log y dy is finite. 2

The procedure in the present paper is the same as that of [4], that is, the proof of Theorem 2.1 is based on the following propositions concerning the Gaussian and the stable laws, respectively, and a Berry-Esseen remainder term type argument.

Proposition 2.1 Let 0 < p < 2, suppose that Y, X1, X2, . . . are i.i.d. standard normal random variables, and set Sn =Pn

k=1Xk, n ≥ 1. Then lim

ε&0

X^∞

n=1

1

nP (|S_n| ≥ εn^1/p) + 2p

2 − plog ε

= 2(1 − p)

2 − p γ − p 2 − plog 2.

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Proposition 2.2 Let 1 ≤ p < α ≤ 2, suppose that Y, X1, X2, . . . are i.i.d. Stable(α)-distributed random variables with mean 0, and set Sn =Pn

ε&0

X^∞

n=1

1

nP (|S_n| ≥ εn^1/p) + αp

α − plog ε

= αp

α − p· E log |Y | + γ.

With arguments similar to those of the proof of Theorem 2.1, we also obtain the following convergence rate statement related to Theorem 1.5.

Theorem 2.2 Let 1 ≤ p < r < α, suppose that X, X1, X2, . . . are i.i.d. random variables with mean 0 and distribution function F belonging to the normal domain of attraction of a nondegenerate stable distribution (function) G with index α ∈ (1, 2], and set S_n=Pn

k=1X_k, n ≥ 1.

(a) If α = 2, Var (X) = σ², and E|X|^2r/p< ∞, then lim

ε&0

X^∞

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p· ε−2(r−p)/(2−p)E|Y |2(r−p)/(2−p)

= −%_r,p, (2.3)

where Y is N (0, σ²)-distributed and %r,p=P∞

n=1n^r/p−2P (Sn= 0).

(b) If 1 < α < 2 andR∞

−∞|x|^αr/p−1|F (x) − G(x)| dx < ∞, then lim

ε&0

X^∞

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) − p

r − p· ε−α(r−p)/(α−p)E|Y |α(r−p)/(α−p)

= −%r,p, (2.4)

where %_r,p is as defined above and Y is Stable(α)-distributed with mean 0.

Remark 2.2 It can easily be verified that %^r,pin Theorem 2.2 is finite under the given assumptions.

Namely, in order to verify this, note that, if α = 2 and E|X|^2r/p< ∞, thenP∞

n=1n^r/p−2∆n< ∞, where ∆n = sup_y

P (|Sn| ≥ n^1/2y) − P (|Y | ≥ y)

(cf. Heyde [7], Theorem, p. 12, with δ = 2r/p − 2).

Moreover, if 1 < α < 2 and R∞

−∞|x|^αr/p−1|F (x) − G(x)| dx < ∞, the latter sum also converges (see Hall [6], pp. 351-352, with β = αr/p).

Now, let x > 0 be fixed. Then

%_r,p =

∞

X

n=1

n^r/p−2P (Sn= 0) ≤

∞

X

n=1

n^r/p−2P (|Sn| < x/n^1/2)

≤

∞

X

n=1

n^r/p−2

P (|Sn| < n^1/2(x/n)) − P (|Y | < x/n) +

∞

X

n=1

n^r/p−2P (|Y | < x/n)

≤

∞

X

n=1

n^r/p−2∆_n+ C

∞

X

n=1

n^r/p−3,

where C is a positive constant. Then the first sum is finite (in view of Hall [6] and Heyde [7]), and the second sum is finite since r/p < 2, which proves the finiteness of %r,p. 2 The proof of Theorem 2.2 is based on the following propositions, again concerning the Gaussian and the stable laws, respectively.

Proposition 2.3 Let 1 ≤ p < r < 2, and suppose that Y, X1, X2, . . . are i.i.d. N (0, σ²)-distributed random variables with σ²> 0, and set Sn =Pn

k=1Xk, n ≥ 1. Then

ε&0lim

X^∞

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) − p

r − p· ε−2(r−p)/(2−p)E|Y |2(r−p)/(2−p)

= 0.

Proposition 2.4 Let 1 ≤ p < r < α < 2, suppose that Y, X1, X2, . . . are i.i.d. Stable(α)-distributed random variables with mean 0, and set Sn =Pn

ε&0

X^∞

n=1

n^r/p−2P (|S_n| ≥ εn^1/p) − p

r − p· ε−α(r−p)/(α−p)E|Y |α(r−p)/(α−p)

= 0.

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3 Proofs

Proof of Proposition 2.1. Let p ∈ (0, 2) and Y be a standard normal random variable. Then, λp(ε) =

∞

X

n=1

1

nP (|Sn| ≥ εn^1/p) =

∞

X

n=1

1

nP (|Y | ≥ εn^(2−p)/(2p))

=

∞

X

j=1

1 j

r2 π

Z ∞ εj^(2−p)/(2p)

exp{−y²/2} dy

= r2

π

∞

X

j=1

1 j

∞

X

n=j

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= r2

π

∞

X

n=1

Xⁿ

j=1

1 j

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= r2

π

∞

X

n=1

(log n + γ_n)

Z ε(n+1)^(2−p)/2p εn^(2−p)/(2p)

exp{−y²/2} dy

= r2

π

∞

X

n=1

(log n + γ)

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

+ r2

π

∞

X

n=1

(γn− γ)

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= Σp(ε) + Rp(ε) . (3.1)

Now, let δ ∈ (0, 1) be small and choose n0 such that |γn− γ| < δ for n > n0. By splitting the sum at n0into two parts and noticing that 0 < γn, γ ≤ 1, we first observe that

lim sup

ε&0

|Rp(ε)| ≤ lim sup

ε&0

P (ε ≤ |Y | < ε(n0+ 1)^{(α−p)/(αp)}) + δP (|Y | ≥ ε(n0+ 1)^{(α−p)/(αp)})

≤ δ.(3.2) Next we note that, as ε & 0,

Σp(ε) = r2

π

∞

X

n=1

log n · exp{−ε²n^(2−p)/p/2} · (ε(n + 1)^(2−p)/(2p)− εn^(2−p)/(2p)) + γ + O(ε)

= ε

√

2π ·2 − p p

∞

X

n=1

log n · n(2−3p)/(2p) 1 + O(1/n) exp{−ε²n^(2−p)/p/2} + γ + O(ε) , (3.3) where the constant in the O(1/n)-term is independent of ε. Furthermore, via a change of variable,

√ε

2π ·2 − p p

Z ∞ 0

log x · x(2−3p)/(2p)

· exp{−ε²x^(2−p)/p/2} dx

= 1

√π· p 2 − p

Z ∞ 0

log y + log 2 − 2 log ε

√y · e^−ydy

= log 2 − 2 log ε

√π · p

2 − p· Γ(1/2) + 1

√π· p 2 − p ·

Z ∞ 0

log y

√y · e^−ydy

= − 2p

2 − plog ε + p

2 − plog 2 + 1

√π· p 2 − p ·

Z ∞ 0

log y

√y · e^−ydy, (3.4)

so that, upon recalling that

√1 π

Z ∞ 0

log y

√y · e^−ydy = Γ⁰(1/2)

Γ(1/2) = −γ − 2 log 2, and by combining the above numbered relations, we finally obtain that

lim

ε&0

λp(ε) + 2p

2 − plog ε

= p

2 − p

log 2 − 2 log 2 + γ

1 − p 2 − p

= γ2(1 − p) 2 − p − p

2 − plog 2 , in view of the arbitrariness of δ.

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It remains to justify that the sum in (3.3) can be approximated by the integral in (3.4). Toward that end, note that in the sum we can always replace log n by log(n + 1) and n(2−3p)/(2p)by (n + 1)(2−3p)/(2p)

(in view of the O(1/n)-term) and thus get either a lower or an upper bound.

Moreover, the remainder term is negligible, i.e., lim

ε&0 ε

∞

X

n=1

log n · n(2−3p)/(2p)· 1

n· exp{−ε²n^(2−p)/p/2}

= lim

ε&0

∞

X

n=1

log n

n² ε²n^(2−p)/p1/2

exp{−ε²n^(2−p)/p/2} = 0,

by dominated convergence. 2

Proof of Proposition 2.2. The proof follows the basic lines of the previous one.

Let Y ∈ Stable(α), set Ψ(y) = P (|Y | ≥ y), y > 0, and note that P (|Y | ≥ y) = −R∞

y dΨ(x). Now, λ_α,p(ε) =

∞

X

n=1

1

nP (|S_n| ≥ εn^1/p) =

∞

X

n=1

1

nP |Y | ≥ εn^{(α−p)/(αp)}

= −

∞

X

j=1

1 j

Z ∞ εj^{(α−p)/(αp)}

dΨ(y) = −

∞

X

j=1

1 j

∞

X

n=j

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= −

∞

X

n=1

Xⁿ

j=1

1 j

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= −

∞

X

n=1

(log n + γn)

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= −

∞

X

n=1

(log n + γ)

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y) −

∞

X

n=1

(γ_n− γ)

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= −Σα,p(ε) − R_α,p(ε), (3.5)

where, again, γ is Euler’s constant.

Once again, let δ ∈ (0, 1) be small and choose n0 such that |γn− γ| < δ for n > n0. Then, by arguing as in (3.2), we first observe that

lim sup

ε&0

|Rα,p(ε)| ≤ δ . Next, noticing that

log n = αp α − p

log εn^{(α−p)/(αp)} − log ε , it follows that

−Σα,p(ε) ≤ − αp α − p

Z ∞ ε

log y dΨ(y) + αp

α − p· log ε − γ · Z ∞

ε

dΨ(y)

= − αp

α − p − E log |Y | − Z ε

0

log y dΨ(y) + αp

α − p· log ε − γ · − 1 − Z ε

0

dΨ(y)

= − αp

α − p · log ε + γ · (1 + O(ε)) + αp

α − p· E log |Y | + O(ε))

= − αp

α − p· log ε + γ + αp

α − p · E log |Y | + O(ε log ε) as ε & 0 , which, by combining as before, results in

lim sup

ε&0

λα,p+ αp

α − p· log ε

≤ αp

α − p · E log |Y | + γ . Upon observing that λ_α,p can similarly be represented as

λα,p= −

∞

X

n=1

{log(n + 1) + ˜γn}

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y),

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where ˜γn = γn+ log n − log(n + 1) → γ as n → ∞, and by arguments analogous to those above, we obtain the same lower bound

lim inf

ε&0

λα,p+ αp

α − p· log ε

≥ αp

α − p · E log |Y | + γ

which completes the proof. 2

Proof of Theorem 2.1 (a). Without loss of generality we can and will assume in the sequel that σ²= 1. Set ∆n= sup_y

P (|Sn| ≥ n^1/2y) − P (|Y | ≥ y)

as before, and

∆n(ε) = P (|Sn| ≥ n^1/pε) − P (|Y | ≥ n^1/p−1/2ε) = P (|Y | < n^1/p−1/2ε) − P (|Sn| < n^1/pε), where Y has a standard normal distribution.

We shall make use of the fact that, if EX²log(1 + |X|) < ∞, then P∞

n=1n⁻¹∆n < ∞ (cf. [7], Theorem, p. 12; in fact, the two assertions are equivalent).

Upon recalling Remark 2.1, let δ > 0 and choose n₀such that

∞

X

n=n0+1

1

n∆_n< δ and

∞

X

n=n0+1

1

nP (S_n= 0) < δ.

Then, splitting the sum at n₀and using λ_p(ε) from the proof of Proposition 2.1, yields

∞

X

n=1

1

nP (|Sn| ≥ εn^1/p) =

∞

X

n=1

1

n∆n(ε) + λp(ε) =

n₀

X

n=1

1

n∆n(ε) +

∞

X

n=n₀+1

1

n∆n(ε) + λp(ε).

Therefore, in view of our choice of n0, the continuity of F , and the fact that |∆n(ε)| ≤ ∆n, we obtain lim sup

ε&0

∞

X

n=1

1

nP (|Sn| ≥ εn^1/p) ≤ −

n0

X

n=1

1

nP (Sn= 0) + δ + λp ≤ λp− % + 2δ, where λp= limε&0λp(ε) from Proposition 2.1, and where % is finite (Remark 2.1 again).

Since δ can be chosen arbitrarily small, we conclude that lim sup

ε&0

∞

X

n=1

1

nP (|Sn| ≥ εn^1/p) ≤ λp− % . A corresponding estimate from below gives

lim inf

ε&0

∞

X

n=1

1

nP (|S_n| ≥ εn^1/p) ≥ λ_p− %, and we are done.

(b) The arguments for the proof of (2.2) are exactly the same as above. Just note that, under the given conditions,P∞

n=1n⁻¹∆_n< ∞ still is in force (cf. Hall [6], pp. 351-352, with β = α), and make use of

Proposition 2.2 instead of Proposition 2.1. 2

Proof of Proposition 2.3. Let 1 ≤ p < r < 2 and Y be a standard normal random variable. Then, by arguments similar to those in the proof of Proposition 2.1,

λr,p(ε) =

∞

X

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) =

∞

X

n=1

n^r/p−2P (|Y | ≥ εn^(2−p)/(2p))

= r2

π

∞

X

n=1

Xⁿ

j=1

j^{(r−p)/p−1}Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= p

r − p r2

π

∞

X

n=1

n^(r−p)/p 1 + O(1/n)

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= p

r − pε⁻^2(r−p)^2−p r2

π

∞

X

n=1

εn^2−p^2p ^2(r−p)_2−p

1 + O(1/n)

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy, (3.6) since Pn

j=1j^{(r−p)/p−1} = _r−p^p n^(r−p)/p 1 + O(1/n), and the constant in the O(1/n)-term is, again, independent of ε.

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Next, by straightforward computations, r2

π

∞

X

n=1

εn^2−p^2p ^2(r−p)_2−p

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= r2

π Z ∞

0

y^2(r−p)^2−p exp{−y²/2} dy + O ε^2(r−p)^2−p ⁺¹

= E|Y |^2(r−p)^2−p + O ε^2(r−p)^2−p ⁺¹. (3.7)

On the other hand,

∞

X

n=1

n^(r−p)/p1 n

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy

= ε⁻^2(r−2p)^2−p

∞

X

n=1

εn^2−p^2p ^2(r−2p)_2−p

Z ε(n+1)^(2−p)/(2p) εn^(2−p)/(2p)

exp{−y²/2} dy. (3.8)

For δ > 0 arbitrarily small, choose N = N (ε) = b(δ/ε)^2p/(2−p)c − 1. Then, by splitting the above sum at N and observing that r < 2p (≥ 2), we get

∞

X

n=1

n^(r−p)/p 1 n

Z ε(n+1)^(2−p)/2p εn^(2−p)/2p

exp{−y²/2} dy ≤ ε(N + 1)^(2−p)/(2p)+ O ε−2(r−2p)/(2−p)

≤ δ + o(1) as ε & 0. (3.9)

In view of the arbitrariness of δ, a combination of (3.6)–(3.9) finishes the proof. 2 Proof of Proposition 2.4. The proof follows the usual pattern. Let 1 ≤ p < r < α < 2, Y ∈

Stable(α) with mean 0, set Ψ(y) = P (|Y | ≥ y), y > 0, and recall that P (|Y | ≥ y) = −R∞ y dΨ(x).

Then,

λα,r,p(ε) =

∞

X

n=1

n^r/p−2P (|Sn| ≥ εn^1/p) =

∞

X

n=1

n^r/p−2P (|Y | ≥ εn^{(α−p)/(αp)})

= −

∞

X

n=1

Xⁿ

j=1

j^{(r−p)/p−1}Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= − p

r − p

∞

X

n=1

n^(r−p)/p 1 + O(1/n)

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y)

= − p

r − p· ε⁻^α(r−p)^α−p

∞

X

n=1

εn^α−p^αp ^α(r−p)_α−p

1 + O(1/n)

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y), (3.10)

where the constant in the O(1/n)-term is, again, independent of ε.

Now, similar to the proof of Proposition 2.4, it can be shown that

−

∞

X

n=1

εn^α−p^αp ^α(r−p)_α−p

Z ε(n+1)^{(α−p)/(αp)} εn^{(α−p)/(αp)}

dΨ(y) = E|Y |α(r−p)/(α−p)+ O ε−α(r−2p)/(α−p)+1, (3.11)

and that the remainder term is negligible. A combination of (3.10) and (3.11) terminates the proof. 2 Proof of Theorem 2.2 (a). The arguments for (2.3) follow the lines of the proof of (2.1). In view of Remark 2.2,P∞

n=1n^r/p−2∆n< ∞, so that we only have to replace λp by λr,p= limε&0λr,p(ε) from Proposition 2.3 and % by %r,p=P∞

n=1n^r/p−2P (Sn = 0), which is finite.

(b) Since again P∞

n=1n^r/p−2∆_n < ∞, the arguments for the proof of (2.4) are exactly the same as

above with Proposition 2.4 replacing Proposition 2.3. 2

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References

[1] Baum, L.E. and Katz, M. (1965). Convergence rates in the law of large numbers. Trans. Amer.

Math. Soc. 120, 108-123.

[2] Chen, R. (1978). A remark on the tail probability of a distribution. J. Multivariate Analysis 8, 328-333.

[3] Gut, A. and Sp˘ataru, A. (2000). Precise asymptotics in the Baum-Katz and Davis law of large numbers. J. Math. Anal. Appl. 248, 233-246.

[4] Gut, A. and Steinebach, J. (2011a). Convergence rates in precise asymptotics. U.U.D.M.

Report 2011:11 (submitted).

[5] Gut, A. and Steinebach, J. (2011b). Precise asymptotics—a general approach (in preparation).

[6] Hall, P. (1981). Two-sided bounds on the rate of convergence to a stable law. Z. Wahrscheinlich- keitstheorie verw. Geb. 57, 349-364.

[7] Heyde, C.C. (1967). On the influence of moments on the rate of convergence to the normal distribution. Z. Wahrscheinlichkeitstheorie verw. Geb. 8, 12-18.

[8] Heyde, C.C. (1975). A supplement to the strong law of large numbers. J. Appl. Probab. 12, 173-175.

[9] Klesov, O.I. (1994). On the convergence rate in a theorem of Heyde. Theory Probab. Math.

Statist. 49, 83-87 (1995); translated from Teor. ˘Imov¯ır. Mat. Stat. 49 (1993), 119-125 (Ukrainian).

[10] Sp˘ataru, A. (1999). Precise asymptotics in Spitzer’s law of large numbers. J. Theoret. Probab.

12, 811-819.

[11] Spitzer, F. (1956). A combinatorial lemma and its applications to probability theory. Trans.

Amer. Math. Soc. 82, 323-339.

[12] Spitzer, F. (1960). A Tauberian theorem and its probability interpretation. Trans. Amer. Math.

Soc. 94, 150-169.

Allan Gut, Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden;

allan.gut@math.uu.se (corresponding author)

Josef Steinebach, Universität zu Köln, Mathematisches Institut, Weyertal 86-90, D-50 931 Köln, Germany;

jost@math.uni-koeln.de