An almost sure limit theorem for moving averages of random variables between the strong law of large numbers and the Erdös-Rényi law

AN ALMOST SURE LIMIT THEOREM FOR MOVING

AVERAGES OF RANDOM VARIABLES BETWEEN THE

STRONG LAW OF LARGE NUMBERS AND THE

ERD

OS-R

ENYI LAW

HARTMUT LANZINGER

Abstract. We prove a strong law of large numbers for moving aver- ages of the form (logⁿ)^{;pPn+(logk =n+1n)p]Xk}when the moment condition

Eexp^ftjX1j1=pgis imposed (with some^p>1). It will turn out that due to the extreme terms among the ^Xk these means do not satisfy a strong law in the classical sense but we can identify its upper and lower limit.

1. Introduction

In this paper we intend to close a gap appearing in the theory of strong limit theorems for moving averages of random variables. To start with we recall the classical strong law of large numbers due to Kolmogoro (1930, 1933):

Theorem KLet

X

,(

X

^k)^{1k =1} be independent, identically distributed random variables.

Then the sequence 1

n

n

X

k =1

X

^k converges almost surely as

n

^!1 if and only if ^Ej

X

^j

<

¹.

In this case the a.s. limit equals

=^E

X

.

One of the many ways to generalize this result involves so called moving averagesof random variables. In this context moving averages are means of the form

b

^{;1n Pn+bnk =n+1}

X

^k with a monotonically increasing sequence (

b

ⁿ)¹ⁿ⁼¹,

b

^{n 2 N} for all

n

^{2 N} such that

b

^{n ! 1} (

n

^{! 1}). For moving av- erages there are well{known analytical results that relate strong laws for these means to strong laws for certain classes of weighted means (partic- ularly summability methods). See e.g. Chow (1973) for such results con- cerning Euler methods, Bingham and Tenenbaum (1986) and Bingham and Goldie (1988)for corresponding theorems on Riesz means. Following this ap- proach one can prove strong laws for certain summability methods if strong laws for suitable moving averages are given. The reader may consult Lai (1974), Chow (1973), Bingham and Tenenbaum (1986), Bingham and Mae- jima (1985) as well as Bingham and Stadtmuller (1990) for a large variety

The results of this paper form part of the author's dissertation written at the University of Ulm, Germany, under the guidance of Professor U. Stadtm uller.

URL address of the journal: http://www.emath.fr/ps/

Received by the journal February 25, 1997. Revised February 18, 1998. Accepted for publication September 17, 1998.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

of results of that kind. Strong laws for very general classes of summabil- ity methods were obtained by Kiesel (1993). Of course this list does not aim at completeness. For a more complete account the reader is referred to Bingham (1985, 1988). The behavior of the sequence

b

^{;1n Pn+bk =n+1n}

X

^k1 diers from

b

^{;1n Pbk =1n}

X

^k1 n=1

n=1

as far as the covariance structures of the se- quences are concerned. This covariance structure is crucial for almost sure convergence whereas from the point of view of convergence in probability there is no dierence at all between both sequences. What can be said in general is that one has to impose the higher moment conditions the more slowly (

b

ⁿ)¹ⁿ⁼¹grows in order to obtain a strong law of large numbers. This can e.g. be seen from the following strong law for moving averages which is implicit in Chow (1973) and then is stated again in a more general framework by Bingham and Tenenbaum (1986):

Theorem C{BT Let(

X

^k)^{1k =1} be a sequence of independent, identically dis- tributed random variables and

p >

1.

Then 1

n

^1=p

n+n 1=p

]

X

k =n+1

X

^{k !}

f

:

s

:

if and only if

Ej

X

^jp

<

¹ and ^E

X

=

:

Similar results hold in situations with more general moment conditions such as^E

(^j

X

^j)

<

¹ with functions

more general than powers but with poly- nomial growth. For a theorem of this kind cf. Bingham and Goldie (1988).

This particular theorem applies to functions like

(

x

) =

x

^p for some

p >

1 but not to

(

x

) =

e

^txfor a

t >

0. It is not very surprising that some condi- tion on the growth of

is needed in order for this result to hold since Shepp (1964) proved the following:

Theorem S Let

X

,(

X

^k)^{1k =1} be independent, identically distributed random variables with ^E

X

= 0 and

M

(

t

) =^E

e

^tX

<

¹ for all

t

in a neighborhood of 0.

Dene

m

(

x

) = sup

t2R

f

xt

^;log

M

(

t

)^g for

x >

0. If

m

(

) = 1

=c

holds for

c

and

then it follows:

limsup

n!1

c

log1

n

n+clogn]

X

k =n+1

X

^k =

a

:

s

:

If

c

varies in a suitable non{degenerate interval then the distribution of

X

is uniquely determined by the limit.

Erdos and A. Renyi (1970) later proved a similar result, obviously unaware of Shepp's work.

Note that if the moment generating function of

X

exists in an open neigh- borhood of 0 then there is a

c

^{0 2R}such that for every

c > c

⁰ we may nd some

^{2 R}with

m

(

) = 1

=c

. Hence this condition is fullled at least for all

c

² (

c

^{0 1}). We may in particular let

c

vary in an interval of positive length.

We will denote such results as Erdos{Renyi{Shepp laws or Erdos{Renyi laws in the following. In contrast to the classical strong laws of large numbers we do not have invariance of the limit in this case. These topics also attracted a number of authors. We want to mention only Csorgo and J. Steinebach (1981), Steinebach (1978), Kiesel and Stadtmuller (1996), Deheuvels and Devroye (1987).

Comparing the above theorems one discovers that none of them applies to moving averages like (log

n

)^{;2Pn+logk =n+12n]}

X

^k under the moment condition

E(exp^fj

X

^j1=2g

<

¹. The function

(

x

) = exp^f

x

^1=2g is not of polynomial growth so the result of Bingham and Goldie does not apply. But on the other hand we do not have any information about the moment generating function ^E

e

^tX that may not exist for any

t

⁶= 0 in spite of the moment condition. So Theorem S does not apply either. Thus we can ask the ques- tion whether anything can be said about the behavior of moving averages like (log

n

)^{;2Pn+logk =n+12n]}

X

^k under the above moment condition, if a classical strong law or an Erdos{Renyi type law holds or if a third possibility applies.

The answer to this question (in a more general form) is the main objective of this work. There are only few papers dealing with such averages. We only want to mention de Acosta and Kuelbs (1983) who partially examined such means in a very general setting (i.e. for random variables taking values in a separable Banach space).

2. Main results

From now on we assume without further comment that

X

, (

X

^k)^{1k =1} are in- dependent, identically distributed random variables. We further set

g

^p(

x

) = sgn

x

^j

x

^j1=p for

p

1 and

x

^2R.

Theorem 2.1. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ = (log

n

)^p. Further let

t

¹

t

²²(0 ¹].

Then the following are equivalent:

(i) ^E

e

^{tg (X)}

<

¹for

t

²(^;

t

¹

t

²),^E

e

^{tg (X)}=¹for

t

^62;

t

¹

t

²] and^E

X

=

.

(ii) liminf

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k =

^;

t

^{1p1 a}

:

s

:

and limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k =

+ 1

t

^{p2 a}

:

s

:

Remark 2.2. Note that there might be dierent triples (

t

¹

t

²) such that the respective values of

+

t

^;p2 and

^;

t

^;p1 agree. Thus the statement of part (ii)⁾(i) of the assertion is to be read as follows:

If liminf^n!1

a

^{;1n Pn+ank =n+1}

X

^k and limsup^n!1

a

^{;1n Pn+ank =n+1}

X

^k both are - nite almost surely then ^E

e

^{tg (X)}

<

¹in a neighborhood of 0. Hence^Ej

X

^j

<

1.

Setting ^E

X

=

we can now write liminf^n!1

a

^{;1n Pn+ak =n+1n}

X

^k as well as limsup^n!1

a

^{;1n Pn+ank =n+1}

X

^k in the form

^;

t

^;p1 and

+

t

^;p2 , respectively, then the moment condition (i) holds.

Similar remarks also apply to all other results of this kind stated here.

Remark 2.3. Observe that the upper limit occuring in part (ii) of the as- sertion equals limsup^n!1

X

ⁿ

=a

ⁿwhich follows easily from the usual Borel{

Cantelli argument. That means that the moving averages contain terms of the size of the norming constants again and again and that the moving av- erages become at most as large as these terms. This suggests that precisely these terms determine the non-classical behavior of these means which will be shown later. In the classical case such terms cannot occur because the moment conditions imposed on

X

or

KX

for arbitrary

K >

0 are equivalent.

Remark 2.4. The special case

t

¹ =

t

² =¹ of Theorem 2.1 yields a strong law in the classical sense.

A similar proof also yields

Theorem 2.5. For some

p >

1 we dene

a

ⁿ= (log

n

)^p. Then the following are equivalent:

(i) ^E

e

^tjXj1=p

<

¹ for some

t >

0, ^E

X

=

. (ii)

lim

n!1

c

1ⁿ

n+c

n

X

k =n+1

X

^k =

for every monotonically increasing sequence(

c

ⁿ)¹ⁿ⁼¹ with

c

ⁿ1 and

c

ⁿ

a

^{n ! 1 (}

n

^!1)

:

In Theorem 2.1 the upper and lower limits might dier from ^E

X

. So we are in a situation that at least resembles the one of the law of the iterated logarithm. Like there one can also ask for the set of limit points of the sequence of moving averages in our setting. This question is answered by the next result. We denote here and later on the set of limit points of a given real sequence (

x

ⁿ)¹ⁿ⁼¹ by^C(^f

x

^ng).

Theorem 2.6. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ = (log

n

)^p. Further let

t

¹

t

²²(0 ¹].

Then the following are equivalent:

(i) ^E

e

^{tg (X)}

<

¹for

t

²(^;

t

¹

t

²),^E

e

^{tg (X)}=¹for

t

^62;

t

¹

t

²] and^E

X

=

.

(ii) ^C

( 1

a

ⁿ

n+a

n

X

k =n+1

X

^k

)!

=

^;

t

^{1p1 + 1}

t

^{p2 a}

:

s

:

We now reconsider an aspect observed above. We have seen that the non- vanishing upper limit in Theorem 2.1 was a consequence of the largest terms occuring. This leads to the idea that it should be possible to prove a strong law in the sense of almost sure convergence to the mean under the moment conditions imposed above for moving averages slightly modied by excluding some terms with large modulus. This is done in Theorem 2.7.

Results of this kind were already shown by Mori (1976, 1977) for the classi- cal strong law of large numbers and by Grin (1988, 1988a) for the law of the iterated logarithm with some ideas being due to Feller (1968). In these classical theorems the moment conditions ^Ej

X

^j

<

¹ or^E

X

²

<

¹, respec- tively, can be weakened by removing extremal terms. The methods we use in the sequel partially rely on techniques developed by Mori and Grin.

Theorem 2.7. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ = (log

n

) . Further let

t

¹

t

²

>

0 be given.

We assume ^E

e

^{tg (X)}

<

¹ for all

t

^2;

t

¹

t

²] and ^E

X

=

.

Finally let (

r

ⁿ)¹ⁿ⁼¹ be a monotonically increasing sequence of positive inte- gers such that

r

^n!1 (

n

^!1). Assume that there exists an

>

1 and a

>

0 with

liminf

n!1

a

ⁿ

r

ⁿ(log

n

)

>

0

:

(2.1) Let

n+a

n

X

k =n+1

X

^k denote the sum ^n+anX

k =n+1

X

^k with the

r

ⁿlargest and the

r

ⁿsmallest terms excluded.

Then 1

a

ⁿ

n+an

X

k =n+1

X

^{k !}

a.s. (

n

^!1)

:

Remark 2.8. Condition (2.1) is satised e.g. by

r

ⁿ = (log

n

) with

² (0

p

).

Theorem 2.9 shows that the condition

r

^n!1 (

n

^!1) of Theorem 2.7 is necessary. So there is no

r

^2Nsuch that the moving averages without the

r

largest and the

r

smallest terms converge to the mean almost surely unless

E

e

^tjXj1=psgnX

<

¹ for all

t

^2Rin which case we have almost sure conver- gence to^E

X

for the original moving averages already without removing any terms.

Theorem 2.9. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ = (log

n

)^p. Further let

r

^2Nas well as

t

⁰

t

¹

t

²

>

0 be given.

We assume ^E

e

^{tg (X)}

<

¹ for

t

^2;

t

¹

t

⁰] and ^E

e

^{t2g (X)}=¹. Finally let

n+a

n

X

k =n+1

X

^k denote the sum ^n+anX

k =n+1

X

^k with the

r

^;1 largest sum- mands removed. Then

limsup

n!1

a

1ⁿ

n+an

X

k =n+1

X

^k

r

^p1

t

^p2

:

Remark 2.10. The proof of Theorem 2.9 shows that the r-1 smallest sum- mands can be removed, too.

3. Auxiliary results First we want to introduce some notation.

x

2

X

k =x

1

means ^X

x

1 k x

2 k 2Z

.

s

^x =

s

^x]is dened the same way for an arbitrary real sequence (

s

ⁿ)¹ⁿ⁼⁰ and an arbitrary

x

0.

The variable

C

is supposed to represent a positive constant that may change within one sqeuence of inequalities.

For

p >

1 we denote by

q

the number

q >

1 with

p

^;1+

q

^;1= 1.

We use the notations

Lx

= max^f1 log

x

^g,

LLx

=

L

(

Lx

) and recur- sively dene

L

⁰

x

=

x

and, for

^{2 N}:

L

x

=

L

(

L

^;1

x

) (for

x

0, respectively). We further write for

>

0:

L

x

= (

Lx

).

First we state a result that can be found in Lai (1974a).

Lemma 3.1. Let(

Y

^k)^{1k =1} and(

Z

^k)^{1k =1} be two sequences of random variables on a probability space ( ^F

P

) such that (

Y

¹

Y

²

::: Y

ⁿ) and

Z

ⁿ are inde- pendent for all

n

^2N. Further let

a b

^2Rand

Z

^n;p!

b

as

n

^!1.

Then the following holds:

a) If limsup

n!1

(

Y

ⁿ+

Z

ⁿ)

a

+

b

thenlimsup

n!1

Y

ⁿ

a

a

:

s

:

b) If liminf

n!1

(

Y

ⁿ+

Z

ⁿ)

a

+

b

thenliminf

n!1

Y

ⁿ

a

a

:

s

:

We will also frequently use the well{known inequalities

1 +

x

+

x

²

2

e

^;jxj

e

^x 1 +

x

+

x

²

2

e

^jxj (3.1)

e

^x 1 +

x

+

x

²

2

H

(

x

) (3.2)

where

H

(

x

) = max^f1

e

^xg or simply

1 +

x

e

^x (3.3)

for

x

^2R.

Next we state some easy technical lemmas.

Lemma 3.2. Let the functions

h

^k: (0 ¹)^!(0 ¹) (

k

= 1 2) be monoton- ically increasing. Let

h

¹(

x

)^!1 (

x

^!1) as well as

h

²(

x

)

(

h

¹(

x

))^{2 !}0 (

x

^!

1).

Then we have for every

y

^2R:

1 +

y h

¹(

x

)

h

2 (x)

exp

yh

²⁽

x

)

h

¹(

x

)

(

x

^!1)

:

Proof. It suces to prove

log

1 +

y h

¹(

x

)

h

2 (x)

;

yh

²⁽

x

)

h

¹(

x

) ^!0 (

x

^!1)

:

But this follows easily by Taylor expansion of the logarithm.

The next lemma essentially contains, as a corollary, the Poisson approxima- tion for sums of Bernoulli distributed random variables.

Lemma 3.3. Let(

p

ⁿ)¹ⁿ⁼¹and(

r

ⁿ)¹ⁿ⁼¹ be two sequences of numbers such that

p

ⁿ²0 1] and

np

^n!0(

n

^!1) as well as

r

^n2Nand

r

²ⁿ

=n

^!0(

n

^!1).

Then

n

X

=rn

n

p

ⁿ(1^;

p

ⁿ)^n; (

np

ⁿ)^rn

r

ⁿ! (

n

^!1)

:

Proof. We know

n

X

=r

n

n

p

ⁿ(1^;

p

ⁿ)^{n; ;}

nr

ⁿ

p

^rnn (1^;

p

ⁿ)^n;rn

n

^rn

r

ⁿ!

1^;

r

ⁿ

n

rn

p

^rnn (1^;

p

ⁿ)ⁿ

:

From Lemma 3.2 we obtain

1^;

r

ⁿ

n

rn

exp

;

r

ⁿ²

n

!1 (

n

^!1) as well as

(1^;

p

ⁿ)ⁿ

e

^{;npn !}1 (

n

^!1)

:

Furthermore

r

ⁿ! (

np

ⁿ)^rn

n

X

=rn

n

p

ⁿ(1^;

p

ⁿ)^{n; Xn}

=rn

r

ⁿ!(

n

^;

r

ⁿ)!

(

n

^;

)!

!

p

^{; rnn} (1^;

p

ⁿ)^n;

n

X

=rn

(

n

^;

r

ⁿ)!

(

n

^;

)!(

^;

r

ⁿ)!

p

^{; rn n}(1^;

p

ⁿ)^n;

= ^n;rXn

=0

n

^;

r

ⁿ

p

ⁿ(1^;

p

ⁿ)^{n; rn;} = 1

:

Remark 3.4. The special case

r

ⁿ =

r

= const can e.g. be found in Mori (1976).

Lemma 3.5. Let

Y

be a random variable dened on a probability space ( ^F

P

). Further let

a

^2Rand

t

¹

t

²

>

0. Let

g

(

x

) =

g

^p(

x

) for

x

^2R. Then

E

e

^{tg (Y)}

<

¹ for all

t

²(^;

t

¹

t

²) if and only if

E

e

^{tg (Y;a)}

<

¹ for all

t

²(^;

t

¹

t

²)

:

Proof. It suces to prove one direction. So assume ^E

e

^{tg (Y)}

<

¹ for all

t

²(^;

t

¹

t

²).

Fix

t

² (0

t

²). If

a

0 then

g

(

Y

^;

a

)

g

(

Y

) and the assertion follows immediately.

If

a <

0 the assertion follows from

Y

^;

a

^;

a

for

Y

0 and for

Y >

0

Y

^;

a

=

Y

+^j

a

^j

Y

^1=p+^j

a

^j1=pp

i.e.

g

(

Y

^;

a

)

g

(

Y

) +

g

(^j

a

^j).

The assertion for negative values of

t

follows similarly.

Because we not only want to deduce a limit theorem from a moment condi- tion but also vice versa we need a result that allows us to do this step. In that respect the following proposition is extremely useful. We use a notation matching the situation in later sections.

Proposition 3.6. Let

g

:^R!Rbe a strictly increasing function such that

g

(

x

)^!1 (

x

^!1) and

g

(

x

)^!;1 (

x

^!;1)

:

For

a

ⁿ=

g

^;1(log

n

) let

a

ⁿ

n

K <

¹ for all

n

^2N

:

Let

limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k

s

⁰ as well as

liminf

n!1

a

1ⁿ

n+an

X

k =n+1

X

^{k ;}

s

¹ with suitable constants

s

⁰

>

0 and

s

¹

>

0.

Then ^Ej

X

^j

<

¹. If ^E

X

= 0 then we have also

E

e

^g(1sX)

<

¹ for all

s > s

⁰ and all

s <

^;

s

¹.

Proof. Set

Y

ⁿ= 1

a

ⁿ

n+an;1

X

k =n+1

X

^k and

Z

ⁿ=

X

^n+an

a

ⁿ . Note that

Z

^n;p!0 (

n

^!

1). Then Lemma 3.1 immediately implies liminf

n!1

Y

^n;

s

¹

:

Hence we obtain

limsup

n!1

X

^n+an

a

^{n = limsupn!1}

1

a

ⁿ

n+a

n

X

k =n+1

X

^k;

Y

ⁿ

!

s

⁰+

s

¹

<

¹

:

Therefore

limsup

n!1

X

ⁿ

n

K

(

s

⁰+

s

¹)

:

Thus for every

x > K

(

s

⁰+

s

¹) we have

P

X

ⁿ

n > x

ⁱ

:

o

:

= 0

:

Because of independence of the events^f

X

ⁿ

> nx

^gwe obtain from the Borel{

Cantelli lemma using the notation

Y

=

X x

^:

1

>

^X1

n=1

P

(

X

ⁿ

> nx

) =^X1

n=1

P

(

Y > n

)

= ^X1

k =1 k

X

n=1

P

(

Y

²(

k k

+ 1])^X1

k =1 Z

fY2(k k +1]g

(

Y

^;1)

dP

Z

fY>0g

(

Y

^;1)

dP

=

EY

^+;

P

(

Y >

0)

:

Thus we have proved ^E

Y

⁺

<

¹ and therefore^E

X

⁺

<

¹. ^E

X

^;

<

¹ can be shown similarly. From now on we assume that^E

X

= 0.

Now we set

Y

ⁿ=

X

ⁿ⁺¹

a

^{n and}

Z

ⁿ= 1

a

ⁿ

X

k =n+2

X

^k. Since ^E

X

exists we obtain for arbirarily small

>

0:

P

(

Z

ⁿ

>

) =

P

a

n

;1

X

k =1

X

^k

> a

ⁿ

!

!0 (

n

^!1) by the weak law of large numbers. Hence Lemma 3.1 gives

limsup

n!1

Y

ⁿ

s

⁰

:

So for every

s > s

⁰:

P

(

X

ⁿ

> sa

ⁿi

:

o

:

) = 0

:

By the Borel{Cantelli lemma

1

>

^X1

n=1

P

(

X

ⁿ

> sa

ⁿ) =^X1

n=1

P

e

^g(Xsn)

> n

= ^X1

k =1 k

X

n=1

P

e

^{g(Xsn) 2}(

k k

+ 1]

1

X

k =1

Z

e

g(^Xs)^{2(k k +1]}

e

^g(Xs);1

dP

^E

e

^g(Xs);1

:

So we have proved ^E

e

^g(Xs)

<

¹ for all

s > s

⁰.

The second assertion^E

e

^g(Xs)

<

¹for all

s <

^;

s

¹ follows applying the same argument to (^;

X

^k)^{1k =1}.

4. Proofs of Theorems 2.1 and 2.6

First we prove the following result which contains one part of the main theorem.

Proposition 4.1. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ= (

Ln

)^p. Further let

t

¹

t

²

>

0 be given such that ^E

e

^{;t1g (X)}

<

¹ and ^E

e

^{t2g (X)}

<

¹. In particular, this implies ^Ej

X

^j

<

¹ and we may assume ^E

X

= 0.

Then

;

t

1^{p1 liminfn!1}

a

¹ⁿ

n+an

X

k =n+1

X

^k limsup

n!1

a

1ⁿ

n+an

X

k =n+1

X

^k

t

^1p2

:

(4.1) Proof. It suces to prove the inequality for the upper limit. Fix

s

²²(0

t

²).

We decompose

X

^k into

X

^k0 =

X

^k1fXk s

;p

2 a

k

g and

X

^k00=

X

^k;

X

^k0. Then

1

X

k =1

P

(

X

^k006= 0) = ^X1

k =1

P

e

^{s2g (X)}

> k

= ^{X1 X1}

P

e

^{s2g (X)2}(

n n

+ 1]^E

e

^{s2g (X)}

<

¹

:

Thus

P

(

X

^k006= 0 i.o.) = 0 and therefore lim

n!1

a

1ⁿ

n+an

X

k =n+1

X

^k00= 0

:

(4.2)

Since 1

a

^n+an

=a

^{n !} 1 as

n

^{! 1} it suces to consider only

n

large enough that for some ~

s

^{2 2}(

s

²

t

²):

s

²

a

^n+an

a

ⁿ

1=q

s

~²

< t

²

:

Now observe that^E

X

^k0 0 for all

k

^2N,^E

X

²

<

¹and^E(

X

²

e

^{~s2g (X)})

<

¹. Then (3.2) yields for

k

=

n

+ 1

::: n

+

a

ⁿ and

t

=

s

^p2

a

^;1=qn :

E

e

^tXk0 1 +

t

²

2^E((

X

⁰)^2k

H

(

tX

^k0))

1 +

t

² 2

E

X

²+^E((

X

⁰)^2k

e

^{tXk0 1fX}k⁰ 0g)

1 +

t

² 2

E

X

²+^E(

X

²exp^f

ts

^;p=q2

a

^1=qk

g

(

X

)^g)

1 +

s

^2p2 2

a

^2=qn

E

X

²+^E(

X

²

e

^{~s2g (X)})expⁿ

O

(

a

^;2=qn )^o where we have used (3.3) in the nal step. Hence we obtain for any

x >

0:

P

n+an

X

k =n+1

X

^k0

> xa

ⁿ

!

e

^{;txan n+anY}

k =n+1 E

e

^tXk0

exp^n;

xs

^p2log

n

+

O

(

a

^1;2=qn )^o

exp^f;

xs

^p2log

n

+

o

(log

n

)^g

which yields a convergent series if

x > s

^;p2 . Since

s

^{2 2}(0

t

²) was arbitrary this proves

limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k0

t

^1p2

:

(4.3) Now the assertion follows from (4.2) and (4.3).

The same proof yields the following variant of Proposition 4.1:

Proposition 4.2. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ= (

Ln

)^p. Let

1.

Let

t

¹

t

²

>

0 be given such that ^E

e

^{;t1g (X)}

<

¹ and ^E

e

^{t2g (X)}

<

¹. In particular, this implies ^Ej

X

^j

<

¹ and we may assume ^E

X

= 0.

Then

;

t

1^{p1 liminfn!1}

a

¹ⁿ

n

]+a

n

X

k =n

]+1

X

^k limsup

n!1

a

1ⁿ

n

]+a

n

X

k =n

]+1

X

^k

t

^1p2

:

(4.4)

Proof. (Theorem 2.1) We rst assume that (i) holds. Then the expected value

=^E

X

exists. Now^E

e

^{tg (X; )}

<

¹for

t

²(^;

t

¹

t

²) and^E

e

^{tg (X; )}=¹ for

t

^62;

t

¹

t

²] by Lemma 3.5. Therefore we may without loss of generality assume that

= 0 since otherwise

X

^k may be replaced by

X

^k;

.

Proposition 4.1 immediately yields limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k

t

^{1p2 a}

:

s

:

Similarly we can prove the corresponding statement for the lower limit. On the other hand we know from Kolmogorov's 0{1 law that the upper and lower limit are constant almost surely. If now

limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k

t

^{1p a}

:

s

:

with some

t > t

² then we obtain from Proposition 3.6 that ^E

e

^{g (sX)} =

E

e

^{s1=pg (X)}

<

¹ for all

s

² (0

t

^p), thus particularly for an

s

with

s

^1=p

> t

² contradicting the assumptions. So we have proved

limsup

n!1

a

1ⁿ

n+a

n

X

k =n+1

X

^k = 1

t

^{p2 a}

:

s

:

and the proof of

liminf

n!1

a

1ⁿ

n+an

X

k =n+1

X

^k =^;1

t

^{p1 a}

:

s

:

is similar.

Now we assume (ii). By Proposition 3.6 ^E

e

^{tg (X)}

<

¹ for all

t

in a neigh- borhood of 0. So ^Ej

X

^j

<

¹ and we denote ^E

X

=

. If we replace

X

^k by

X

^{k ;}

again, if necessary, we may assume without loss of generality that

= 0.

Then Proposition 3.6 implies ^E

e

^{tg (X)}

<

¹ for all

t

²(^;

t

¹

t

²). If ^E

e

^{tg (X)}

<

¹ for a

t > t

² then as in the rst part of the proof limsup

n!1

a

1ⁿ

n+an

X

k =n+1

X

^k

t

^{1p a}

:

s

:

would follow from Proposition 4.1 and this would contradict the assump- tions. The assumption ^E

e

^{tg (X)}

<

¹ for some

t <

^;

t

¹ similarly leads to a contradiction.

Remark 4.3. In the special case where an asymptotic for the tail of the distribution function of

X

is known Theorem 2.1 can also be proved using a large deviations result implicit in Nagaev (1979) (this can also be found with a short proof in Gantert (1996)).

Proof. (Theorem 2.6) That (i) follows from (ii) is obvious from Theorem 2.1.

So we assume (i). Using Proposition 4.2 we can show exactly as in the proof of Theorem 2.1 that the moment condition (i) is equivalent to

liminf

n!1

a

1ⁿ

n

]+a

n

X

k =n

]+1

X

^k =

^;

t

^{1p1 a}

:

s

:

and

limsup

n!1

a

1ⁿ

n

]+a

n

X

k =n

]+1

X

^k =

+ 1

t

^{p2 a}

:

s

:

Since

1 is arbitrary we nd for every

s

^2h

^;

t

^;p1 +

t

^{;p2 i} a subse- quence with an almost sure upper or lower limit that equals

s

. The re- mainder follows from a standard argument: If we choose a countable dense subset

S

of ^h

^;

t

^;p1 +

t

^{;p2 i} it follows by excluding the exeptional sets corresponding to elements of

S

:

C ( 1

a

ⁿ

n+an

X

k =n+1

X

^k

)!

S

a.s.

Since the set of limit points must be closed we obtain

C ( 1

a

ⁿ

n+an

X

k =n+1

X

^k

)!

^;

t

^{1p1 + 1}

t

^{p2 a}

:

s

:

The reverse inclusion

C ( 1

a

ⁿ

n+an

X

k =n+1

X

^k

)!

^;

t

^{1p1 + 1}

t

^{p2 a}

:

s

:

is an obvious consequence of Theorem 2.1. This completes the proof.

5. Proofs of Theorems 2.7 and 2.9

Proposition 5.1. For some

p >

1 we dene

g

(

x

) =

g

^p(

x

) and

a

ⁿ =

L

^p

n

. Further let

t

¹

t

²

>

0 be given.

We assume ^E

e

^{tg (X)}

<

¹ for all

t

^2;

t

¹

t

²] and ^E

X

= 0.

Finally let (

u

ⁿ)¹ⁿ⁼¹ and(

l

ⁿ)¹ⁿ⁼¹ be monotonically decreasing sequences of re- als with 0

l

ⁿ

u

ⁿ1,

a

ⁿ

u

^{n !1}

a

ⁿ

l

^n!1 (

n

^!1) and

u

= lim

n!1

u

ⁿ and

l

= lim

n!1

l

ⁿ. Set

U

ⁿ= ^n+anX

k =n+1

X

^k1

; ln

t p

1 anX

k

un

t p

2 an

:

Then

;

l

^1=q

t

^{p1 liminfn!1}

a

¹ⁿ

U

ⁿlimsup

n!1

a

1ⁿ

U

ⁿ

u

^1=q

t

^p2

:

Proof. Set

Y

^k(n)=

X

^k1f;ln

t

;p

1 a

n X

k u

n t

;p

2 a

ng and

ⁿ = ^E

Y

^k(n). Observe that

^{n !} 0 as

n

^{! 1}. Choose

>

0 and

s

^{2 2} (0

t

²) arbitrarily but