Proofs fo r Chapter 5.

In document Sums of independent random variables and regular variation (Page 112-118)

Here X is any r.v with distribution the same as X^ By the Borel-

4. Proofs fo r Chapter 5.

Proof o f Theorem

1. Assume first that limsup S /B(n) ^

n n

2 -1 -2

c[b+ (A)+e] [b+ (A)) a.s., where e > 0 and A > 1. Looking at the proof of Lemma 2 on page 722 of Kesten (1972), we can see that

limsup S. / B(A.) < p a.s. implies limsup S. / B (A .) < p+e a.s.

i

V

i

'

j “

j

V

i '

J

and

J H '

J

"J + l

this implies E P [S _> pB(A,)] < + °°. Here A, - [A^], the greatest

J>1

j

J

J

integer less than or equal to A^ ; Kesten actually gives his result for the sequence A^ * 2^, i.e., he considers A = 2, but his proof works in the above generality, as is easily seen (he also considers only the case A ■ 0 in (1.2), but again his proof is easily modified). Now limsup S /B(n)£p

n n

a.8. implies limsup S /ß(A ) <_ p a.s., so j A1+1 / J+i

J + l

S S

Xj+1 Xj+1 B(Xj+l) , B(A[Aj ]+l) B( [ A^ ]+l) limsup -R7t-y = limsup --rH-- r ~ ^ 7 T ~ T ~ 1 P limsup — ^ ™ --- *

j B(Xj} j B(Xj+l} B(Xj} j B([AJ ] + 1) B([AJ ]) < p b+ (A)b+ 1 p b+ (A) u > B ■ j

by (1.1), remembering also that limsup B(n-»1)/B(n) £ b = b (1+) b (X) ,

n + +

and noting that, under (1.2), limsup S /B(n) >_ 0 a.s.; in fact, for any

n n

6 > 0, P(S /B(n) _> -6 io) > limsup P(S >_ -6B(n)) limsup P(S ^_-A) j^q > 0.

n n n n n

Now under our initial assumption we can conclude from Kesten's modified

2

-1

result that E P[S. _> c(b,(X) + e) B(X.)] < + «. Our proof proceeds

J j V i

from this. For j ^ 1 and n X , S. = S + E X . , so by (1.2) j 1 XJ+1 n isBn+1 i

and Petrov (1975 Theorem 3),

P[S. >_c(b^(A)+e) *B(A ) ] qP | max S >c(b?(A)+e) 1

j+1

J

Ll£ni X4J.1

]

> qP[Sn > (c+e)(b^(A)+e)_:LB(Aj+1)]

■l£n<Aj + i

B(Aj+1) + A

where j is large enough for A/B(Xj) _< e. (Note that, although we have only assumed (1.2) holds for n >_ n Q> it must then hold for n 1, since min S , > - 00 a.s. if X is a proper r.v.). Suppose also that n ^ X . Then we have A^+ ^ “ [A^+ h < A^+ ^ < A[A^] + A <_A(n+1), so

B(XH'1) . B(A(n+l)) B(n-t-l) 2 . .

B(n) - B(n + 1) B(n) - b+ (A) i

for j _> jo (e). Thus for X^ < n we ^ave

P[S. > c(bJ(A) + e)B(A )] > qP( S > <c + e)B(n))

Aj + i + J

+1

n

and adding this from n = X^ + 1 to X^+ ^ gives

(Xj+l)“ 1 (Xj+1-X.)P[Sx > c(b^(X)+e)B(XJ+1)] > lj+l -1 >_ q E n P (S _> (c+e)B(n)) . n=Aj+1 -1 < x ~ i

Since (X^+l) £ X J , adding over j _> j Q gives

« > X E P[S. >c(b?(X)+e)B(X )] >. E n_1P [S > (c+e) B (n) J

which is the required result. By choosing

A near to

1

and

e near to

0,

our hypothesis becomes limsup Sn /B(n)

c/b*1 a.s., and

thus we complete the first half of the proof.

To prove the converse, assume E n ^P[S > cB(n)] < + “ . Take n>l 11

A > 1 and A^ = [A^]; then for j >_ 1,

xj+i , , V i

Z

n

P [ s n

> cB(n)] > X *

Z

P [ S „ ^ c B ( n ) ] n=Xj+l P[Sn > oBCrx)]

X

3

+l<n<X

.+1 n=Xj+l

By Petrov (3-975 Theorem 3) and (1.2), for n _> A^

P [S _> cB (n) ] > q P [ max S , >_ cB (n) + A] > q P [S, 21 (c+e)B(n) ]

" l<J<n 3 Aj

where e > 0 and n is so large that A <_ eB(n). Also, for n £ A and j 1 jQ U ) *

B(n) B(X1+1) B(X[X3 ]+1) B([X3 ]+1) 2...

U X O - » ( x p - B ( [ xj ] + i )

b

([x3 ]) - +( >

'

so that

P[Sn > cB(n) ] > q P [ S Ä > c(b+(A) + e)B(Aj)] for Aj _< n

<_

Aj+ ^, and we have

j+1

inf P[S > cB(n) ] > q P [S. > c(bf(A) 4- e)B(A . ) ]

a . n — a .

xP*Vi

j

leading to, j+1 £ n 1P[S > cB(n)] > q(l-A

1-\

j 1)P[S, > c(b?(A)+e)B(A.)] , ,. n a . T j n=Aj+l j

Summing this over j >_ jQ gives

(4.1) “ >

Z

n

XP[S _> cB(n) ] > n>X. +1 n - 3 -1 -Jo"1 > q(l-X -X )

Z

P[:i

> c(b‘ (X) +

e)B(X.)]

3^ o 3

which means limsup S, /B(A ) <_ cb,(A) a.s., and (by (1.2)) that c 0.

j Aj 3

We now want to see what happens to (S, - S )/B(A ). Note that,

j+1 j J

since [x] <_ x < [x] + 1 for any real x, we have [ ] < A3*'*' - A3 + 1 < [A3+1 - A3 ] + 2, and [A3 +1]-[A3 ] > A3+1 - A3 - 1 >_ [A3+1 - A3 ] - 1. Define A to be those A e (1,2] for which -log (A-l)/log A = k(A), an integer; note that 1 is a limit point of A. Clearly A (A-l) = 1,

k = k(A) >_ 0 and k -*• + 00 as A -*• 1+ through A. Now since A^+ ^ - A^ * * A^(A-l) = A^ ^ < [A3 + 1, similarly A^+ ^ - A^ [A^ ^ ] , from the

above argument we can conclude [A3+^] - [A^] = [A3 + z, where z is an integer between -2 and 1. This means has the same distri­ bution as S

V k +Z

j+i

-j

when A e A. Now we modify the a.s. stability criterion (Loeve (1963 page 252)). Let

J. = sup (S - S, ), so that for x > 0,

1 V n.Vl »

j

P[U. >_ xB(A )] = P[ sup E X >_ xB(A )]

J 3 A .<n<A.,- i=A,+l 3 j “ J+1 j n+A j * p [ sup E X. >_ xB(A ) ]

l<.n£Aj+1-Aj i=Aj+l

3

- P[ sup S<j) > xB(A )] ^ j + l - ^ j n ' j+1 j q ^ P l s p * , > xB(X ) - A] j+1 j J

by Petrov s (1975 Theorem 3) generalisation of Levy s inequality, when n+A

a ) . (1.2) holds, where S

j n

E X. * E X, is the sum of n inde- i-A.+l 1 i-1 Aj+1

J

(j)

pendent random variables. Now note that S^J = S, -S, , and if we j+1 J j+1 j

(4.2) q

Z

P[Uj>xB(X j)] < I P[SA - >(x-e)B(X )]

- Z P[SX

+z I (x-e)B(X

)]

j>3o Aj-k+z J+1

since S. -S. has the same distribution as S. . . Now if z < 0,

V i xj xj-k+z

we can bound P[S

x . , + z — (x-e)B(X

j-k

Lj+1)] above by P[SX _> (x-e)B(X^+1>-A] j~k

by Petrov's (1975) result; but if z > 0 we have to use (1.3). Suppose z = 1; then

P[S > (x-e)B(XJ+1)] < P[8 > (x-2e)B(Xj+1)] +

j“k j~k

+ P [ X X

+ z I e B ( X J+1 )]

J”K

while if z ■ 2, we can iterate this inequality and just add another term P[XX +z >_ eB(A ) ] to the right hand Bide. By (1.3),

j"k Z J

limsup X. /B(A _) <_ limsup X 1 0 a.s., so by the

Aj_ktz j^i Aj — j **■

Borel-Cantelli Lemma, E P [X^ +z >_ eB(A,+ ^)] < + °° for every e > 0.

i ? J 0

J-* z 3

Also, note that

A

*k k+1

liminf B(XJ+1)/B(Xj_k) > liminf B([XJ * A* ])/B(AJ k )

>_ b_(Xk+1) = b_(X(X-l)_1) when A e A, so when x - 2 e >_ c b^(A) /b_(X (A-l) ^) ,

E P[S. > (x-2e)B(A. )] < E P[S. >_ (x-2e)b (X(X-1)"1)B(X )]

j>jrt J-k J J>J +k j j

< 4 - 0 0

by (4.1). Thus we have shown that E P[S

X k+. - < + " when x-2e >_ cb^(X)/b_(X(X-l) 3) , so by (4.2), and the Borel-Cantelll

Lemma , limsup U /B(X ) <_ cb^(A)/b (X(X-l) ^) a.s. But then, for

j j J+1 +

l i m s u p <_ limsup n n <S n - \ >

B< V >

B ( V i > B(n) + limsup

B(X.)

< limsup

S(AJ+1)

B(X1+1>

h

i w + U “sup b ö~7

c b B (\) £ --- + cb (A) a.s., if a e A, b ( M A - l ) “ 1 ) B(A ) ~B~(n)

and l e t t i n g A -* 1+ t h r o u g h A g ives the r e q u i r e d result.

R e m a r k s .

(i)

The a b o v e proof, apart from the use of K e s t e n 1s Lemma, is a f o r m a l i s a t i o n of the m e t h o d s by w h i c h the v a r i o u s v e r s i o n s of the law

of the i terated l o g a r i t h m h a v e al w a y s b e e n proved; n otably, the u s e of a

g e o m e t r i c s u b s e q u e n c e and then the " f i l l i n g in" of the r e m a i n i n g v a l u e s by

the use of Le v y ' s i n e q u a l i t y and the a.s. s t a b i l i t y arg u m e n t in some form.

(ii) In the second half of the p roof of T h e o r e m 1, the a s s u m p t i o n

(1.2) can o b v i o u s l y be w e a k e n e d to

P ( S n >_ - A n ) _> q > 0 for some n o n - d e c r e a s i n g >_ 0, A ^ * o ( B n ) ,

w h e n n > n , — o

(iii) Le t ' s c hange our a s s u m p t i o n s and s u ppose are independent

but not necessarily identically distributed, and symmetric. T h e second

half of the proof of T h e o r e m 1 n o w works, except for the r e q u i r e m e n t that

S. -S h a v e the same d i s t r i b u t i o n as S, . . H o wever, n o t e that

J + l

J

j + 1 j

(S. — S . ) is a seq u e n c e of i n d e p e n d e n t r.v.'s w i t h j+l j

l i msup j I y/ß(A <_ 2cb^(A) a.s., since we showed

j-H-oo

j + l

J

l i m s u p

|s, I /

B(A ) < cb^(A) a.s. Thus, by the B o r e l - C a n t e l l i Lemma, the

j-*~ j ' J

2

-1

s eries in (4.2) conv e r g e s for x > 2cb+ (A)/b_(A(A-l) ), giv i n g at o n c e :

s u p p o s e f u r t h e r that B(n) s a t i s f i e s (1.1) and that (1.3) holds. T h e n

I

n [

I

S

I >_

cB(n)] < +

«

i m plies limsup

|s

|/B(n) 2cb+ (l+b^/b_) a.s.

n>_l n n-H*» n

with when EX. < + as we showed at the end of Section 1.

J

Proof of Corollaries 1 and

2

to Theorem 1:

these are obvious from

In document Sums of independent random variables and regular variation (Page 112-118)