Indian Statistical Institute
Tail Behaviour of Distributions in the Domain of Partial Attraction and Some Related Iterated Logarithm Laws Author(s): G. Divanji and R. Vasudeva Source: Sankhyā: The Indian Journal of Statistics, Series A (1961-2002), Vol. 51, No. 2 (Jun., 1989), pp. 196-204 Published by: Springer on behalf of the Indian Statistical Institute Stable URL: http://www.jstor.org/stable/25050737 . Accessed: 08/08/2013 06:58 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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: The Sankhy? 1989, Volume
Indian Journal of Statistics Series A, Pt. 2, pp. 196-204.
51,
TAIL BEHAVIOUROF DISTRIBUTIONSINTHE DOMAIN OF PARTIAL ATTRACTION AND SOME RELATED ITERATED LOGARITHM LAWS and R. VASUDEVA
By G. DIVANJI*
SUMMARY. i.i.d.
random
iff there
attraction to
a non
the
an
exists
a distribution
be
common
the
with
variable.
of F
tail
and
(nj) such
iterated
obtain
1.
F
F.
Under
sum sequence of (Sn) be a partial is said to be in the domain of partial
let
and
function
distribution
sequence
integer
random
degenerate
characterize
F
Let
variables
India
of Mysore,
University
that
certain
(Sn,),
on
assumptions laws
logarithm
for
normalized,
properly
(Sn) and
converges
sequence (nj) we max !?&!).
the (
Introduction
distributed of independent (i.i.d.) (Xn) be a sequence identically a over common variables defined space (?Q,&, P) (r.v.) probability n 2 Xj, n > 1. Let F denote the distribution let Sn ? function (d.f.)
Let random and
i=i
of
(nj) be an integer
Let
of Xv
(B? ?? oo as
constants
and
subsequence
j -?
Setj ZM =
oo).i
cides with
the sequence of natural if (Zn) converges weakly,
numbers then
and (an )
be sequences (Bn ) . When
B~1SM n. ?aM
nj
nj
and
let
x(nA J'
coin
nj
nj
selection of (n), for proper it is wellknown that the limit
(an) law
(Bn), is stable (or possibly For some subsequence (nj) and for proper degenerate). the limit law is if then selection of (a ) and (Zn 1 ) converges weakly, (B%3 ), j and Kolmo law (see, ex. Gnedenko to be an infinitely known divisible (i) nj < (1972) considered sequences (nj) satisfying Kruglov = the class It r( > 1), and characterized J 5* 15 and (ii) lim or Uj+1jnj
gorov
(1954)).
nj+i> of
j?
all
found
infinitely that
the
be
noted
It may
if
particular, Research C.S.I.R.S.R. AMS Key varying
members that
lim nj+Juj supported
have
of U
the
by
class =
of
all
1, Kruglov
University
are
which
distributions
divisible
of
many stable
laws of
(Zn ).
He
of stable laws. properties laws is included in ??. In
(1972) Mysore
limit
established Junior
Research
that
(i)
Fellowship
the and
Fellowship. subject
(1980) words
and
phrases
classification: : Domain
Primary of partial
60F15, attraction,
Secondary Iterated
60E99. logarithm
functions.
This content downloaded from 14.139.155.135 on Thu, 8 Aug 2013 06:58:02 AM All use subject to JSTOR Terms and Conditions
laws,
Slowly
197
ITERATED LOGARITHM LAWS limit
is a stable (Zn )
law of
law and
(ii) the
sequence
norma
(Zn), properly
to the same stable law. the sub converge Consequently, our are of under interest those sequences Kruglov's setup subsequences (uj) 1. Here Km Uj+1fnj = r, r > with limit has characterized the Kruglov itself
will
lized,
G as either
distribution the
characteristic
normal
or as an
