of Chapter 3, liminf H(xA)/H(x) > bA x-H-oo

In document Sums of independent random variables and regular variation (Page 45-51)

Theorem 1 of Chapter 3, liminf H(xA)/H(x) >_ bA x-H-oo

2b-2

for A > 1, where

V(xX) \ 2 V(xX) H(xA) x ^ U x ) . , 9 - 1 , , 2b-2

»W

w " -

b

abc

lX2h

so lim liminf V(xA)/V(x) = + 00 , which is (2.11); a < 2 as before. So

A-H-oo

far we have that (1.15), (1.16), (1.18) and (2.11) are equivalent and we

need only deal with (1.17). Note first that (2.11) is equivalent to

(2.12) F e S C ( a ) , 0 < a < 2, and lim limsup V(xA)/V(x) = 0.

A-KH- xr-H-°°

Let (2.12) hold and (1.17) fail, so there is a sequence n ’ + 00 for

sn »

which --- A , a r.v. with non-zero normal component; but then

B I n

n l A _ f 1i

u dT(u)]/[a2 -

V(ABn ,)/V(Bn ,) + [a2 _ u dT(u)] > 0 for A > 0,

'

o

which contradicts (2.12). Thus (2.12) implies (1.17). Conversely, let

(1.17) hold, and suppose (2.12) fails, so there is a sequence x^

-*

+ 00

for which liminf V(x A)/V(x ) > e > 0. We can then find k with

q n n n

B^ ^ x^ for which liminf V(Bk A)/V(B^ ) e6 > 0; but there is a

n Sk * n n

k'

<=■

k for which ^- - A, . -> a r.v. with zero normal component, so

n — n B. , k

k' n

V(Bk ,A)/V(Bk ,)

n n

u2dT(u) — + 0, giving a contradiction. Thus

(2.12) holds and the proof of Theorem 6 is complete.

Proof of Theo r e m 7.

Let F e SC and let 1 be a subsequential

limit of F. Suppose first that F, and hence I, are symmetric. Let

have distribution I, let X ^ , X^2\ . . . X ^ be k independent copies

of X-j., and let

s^k) = X ^ + X^2) + ... + x ^

Now there is a sequence n, for which X T = lim S /B , where

i I i n t

S = X, + X„ + ... + X , the X . ’s are distributed as F, and

B = n n i

B

2 (k)

. I u dF(u). Hence we can write X^: 1 -B

n . l

lim S ^ ^ / B , in an i n i n i

obvious notation, where all the component X ^ ’s have distribution F and are independent. Thus we can say, for each i,

(k) + ... +

Si n i n i___________+ Yk

C, B C, i

k n^ k

k P

where is any sequence and for each k, Y^ — *■ 0 as i -*■ + 00 . Now let k" be any sequence. For each k" we can find iQ = i^Ck") so

I k " . -k"

that i iQ implies P(|Y^ | > e) , where e > 0. Hence also I k" I -k" k" P

P ( IYi I > e ) <_ 2 , so -* 0 as k" -+ + 00 . Now consider

o v o v

s (1) + s (2> + ... + s (k,,)

s k )

n i n i n i

_1________ 2_____ 2___________2_ + Yk

Ck„ - B Ck„ io (k") •

The first term on the right hand side has distribution [nio (k")+k" ]*

F (x B C. ,,) , and because of the stochastic compactness of

ni

k

o

Fn (x B ), we can find a subsequence k' of k" for which [ni (k')+ k ']*

F ° (x B C. ,) converges to a non-degenerate distribution, n i (k*)

o

provided we choose C. , so that B C

1 = Bn . This means, K n i (k*) o i (k')+k' o since Yk k 0 as k' -► + TO

V k )

( V

, that converges to a non-

degenerate random variable, and shows that I e SC. For the non-symmetric case, the above proof works for S<k)s, the symmetrised sum, and then from Feller (1971 Lemma 1 page 149), letting be a median of _ (k)J , we have, for x > 0,

2 P(|Sjk)-Mk | > 2 Ck x) < P ( | S ^ S |(k) s > 2 C k x) 1 4 P ( | S ^ J - M j > Ck x) ,(k)

showing that

q (k) M si

is SC, and completing the proof of (1.19).

We now prove (1.20). Let F e SC(a) 0 D^(2), where a < 2, and Sn'

suppose —— - A , XT, where XT has the inf. div. distribution I ;

B . n I I

n

I has canonical components a2 (= 0, by Theorem 6) and T(x). We show that I e SC(a)

0

D p (2) . We have for x > 0 and X 1,

(2.13) T(Xx)/T(x) = lim n'H(AxB ,)/n'H(xB ,) <_ limsup H(Xy)/H(y)

■ n n

< c'A y

-a

by Theorem 4, and, defining W(x)

X > 1,

u dT(u), we have for x > 0 and

W(Ax)/W(x) = lim V(XxBn , )n ' B^^/v(xBn , )n' B^^ <_ limsup V(yX)/V(y)£

n' n n / n y

< c X2-a by Theorem 2. Also for x > 0, since F £ D (2),

(2.14) x2T(x)/W(x) = lim (xB ,)2H(xB ,)/V(xB ,) >

, n n n —

n

_> liminf y2H(y)/V(y) >_ a > 0 , y

and, from the proof of Theorem 1 of Chapter 3, since liminf H(xX)/H(x) x-H-«°

2b““ 2

_> bX for X >_ 1, where b = a/(a+l) > 0, we have for x > 0 and X > 1,

(2.15) T(Xx)/T(x) lim n ’H(XxB ,)/n'H(xB f) liminf H(yX)/H(y)

t n n

> bX y 2b-2

aim to show limsup

P(|x | > xX)/P(|x | > x)

_< c"A a for

X >

1

■v X X

Letting be the symmetrised sum, we have from page 149 of Feller (1971), for x > 0,

P (IXT I > x) = lim P (I S ,-A , B , I > xB ,) >

\

liminf P(|sS , I > 2 x B ,)

'I' , 1 n n n n — 2 , 1 n 1 n

n n

^

liminf n'P(|xS | > 2 xB^,) exp [-n' P ( | Xs | > 2 x B ^ ,) ] .

Now is also SC, since it has the representation

ss ,

n

V

V 1 K ' \ B , A n ’ v n ’ ; ^ B , n A n' X I - X ' ,

and the latter r.v. is inf. div. with canonical tail 2T(x), as is easily seen. Thus

n'P(|xS|

> 2 x B n ,) -* 2T(x) , and so

(2.16) PdXjl > x) > ^ T(2x)e-2T(x)

> i b 22b-2 T(x)e-2T(X> ;

T T

Also for x > 0, define the truncated r.v.'s X^ by : X^ = X^ if IxJ < xB , xf = 0 otherwise. Then if ST = X? + ... + XT ,

1 i 1 — n i n 1 n

P (IS -A B I > xB ) < P(|ST-A B I >xB ) + nP(|x| > xB ) . 1 n n n 1 n — 1 n n n 1 n 1 1 n Now we may choose the centering constants A as A = nB

n n n

-1

udF(u)+0(1)

(where 0(1) depends on x), and then we write by Chebyshev's inequality,

P(|ST - A B I > xB ) < x~2B_2E(ST - A B )2 1 n n n 1 n — n n n n — 9 -2 T -2 -2 2 T = x B VAR(S ) + x B E (S - A B ) n n n n n n = x2b” 2 nVAR(X^) + o(l) , n i since E(S - A B ) = n

n n n udF(u) - AnBn = 0(1). Hence we have, noting xB

r n

that VAR(X^) u dF(u) - udF(u)

-xB

[1 + o(l)]V(xBn) ,

P(|Sn " AnBn l

> xBn ) - X 2ßn 2 n v (xBn> + n P

(|x|

> x

Br)

+ o

(1) ,

and so, going through the subsequence n' gives

(2.17) P (IXT I > x) = lim P (IS , - A ,B ,1 > xB ,)

I , 1 n n n 1 a

n

£ x_2W(x) + T(x) .

This means, from (2.16) and (2.17), for x > 0 and X > 1,

P ( jX lj >

XX) , ,-l,3-2b.-2 W(Xx) W(x) 2T(x) - «<*> x2T(x) , .-l,3-2b T(Xx) 2T(x)

+ b 2

TÖO 6

* K-l03-2b.-2 ,2-a -1 2T(x) , ,-l03-2b , -a 2T(x) < b 2 X c X a e + b 2 c X e + c"X-a

as x , using (2.13) , (2.14) and the fact that T(x) 0 as x -*■ + 00 . From Theorem 4 we will have IeSC(a) H D^(2) if I ^D (2); but by (2.15),

(2.16) and (2.17),

PdXjl >xX) 2b_3 T(Xx) -2T(Xx) x2T(x) . , , 2b-2 > X ) - ^ W(x)+x2T(x) - C°nSt-

as x + + 00 . Thus from Theorem 1 of Chapter 3, I $ D (2) . We now show that F e SC(a) f) Dp (2) implies (1.21). From (2.13) with x = 1, and

(2.14), for X > 1,

P (IXx I > X) < X 2W(X) + T(X) £ a 1 (a+l)T(X) <_ b ic ,T(l)X-1 < h 1

-a

< b V cX a = c"X-a by (2.16) and the fact that

T (1) = lim n'H(B ,) = lim B2 ,H(B ,)/V(B ,) 1 limsup y 2H(y)/V(y) £ c

n n

This proves (1.21), and we now show the converse; let (1.21) hold - we have to show F e SC(cx) C \ D^(2) . Note that I itself is not the normal distribution, so F | D^(2), and we have postulated F e SC, so by

2

Theorem 2 there is a k > 0 for which limsup x H(x)/V(x) < k < + 00 . x-H-°°

Reasoning as above then tells us that 0 < a x T(x)/W(x) _<_ k < +00 for every x > 0. (I has zero normal component by Theorem 6).

Furthermore, from (2.14), (2.17), and (2.16), for

X

> 1,

(2.18) £ a 1 (k+l) p( 1 1 const. X 2 P (|Xp| >

X)

£ const. X

2-a

since we are told that

P

(|

X^.

| > X) <_ c"X , and

P(|x | > 1) > 0

since

I

has infinite variance. Now we show F e SC(a). Suppose not, so that

(2.19) Xa 2 limsup V(xX)/V(x) -*•+«> as X -► + 00 . x

We can find an x - + + 00 for which Xa 2 liminf V(x X)/V(x ) -* + 00 as

n n n n

X -♦■ + » as follows :given e > 0 and any X ^ l + °° with X^+ ^ ^ X^ , choose i (e) so that XJ .. < (l+e)X, for i > i (e). Given a large

o i+1 — i — o

ot-2

A > 0, choose X (A) so large that X limsup V(xX)/V(x) A whenever

0 x

X _> X . For each i there is then a sequence xn (i) for which

In document Sums of independent random variables and regular variation (Page 45-51)