Vo
l.
32
( 2012 )
NO.6
数学杂志
J.
of
Math. (PRC)
A
WEAK
CONVERGENCE
THEOREM
FOR
FUNCTIONAL
PRODUCTS
OF
TRIMMED
SUMS
OF
I.I.
D.
RANDOM
VARIABLES
ZHENG
Fa-mei
(8
伽
olofM
α
them
Abstract:
In this article,
we
study the
weak
invariance principle.
By
using the idea of
self-
normalized limit theorem to trimmed sum,
we
get a
weak
convergence theorem. Meanwhile,
we
extend the
weak
convergence theorem to U-statistics.
Keywords:
weak
invariance principlej the trimmed
sumj
medium tailj
weak
convergence
theoremj U-statistics
2010
MR
Subject
Classification:
60F05
Document
code: A
Article
ID: 0255-7797(2012)06-0977-11
1
Introduction
and
Main
Results
In
the
past
years
,
the
asymptotic
behaviors
of
the
products
of
sums
of
various
random
variables were widely
studied.
Arnold
and
Villaseñor
[1]
considered
sums
of
records
and
ob-
tained
the
following
form
of
the
centrallimit
theorem
(CLT) for
independent
and
identically
distributed
(i.i.d.)
exponential
random
variables
with
the
mean
equal
to
one
,
Z~=llog
Sk
-
nlog(n)
+n
d
川一一
--一→
Jvasn-
→ 00
\/2η
、、，，
J
唱
EA
-
ti
，，，，‘、
Here
and
in
the
sequel, N is a
standard
normal
random
variable
,
and
~
(鸟，写)
stands
for convergence
in
distribution
(in
probability
,
almost
surely).
Observe
that
,
via
the
Stirling
formula
,
relation
(1.
1)
can
be
equivalently
stated
创
(杠)古主产
(1.
2)
Fu
rther
,
Rempala
and
Wesolowski
[10]
removed
the
condition
that
the
distribution
is ex-
ponential
and
showed
the
asymptotic
behavior
of
products
of
partial
sums
holds for
any
sequence
of
i.i.d.
positive
random
variables.
Namely
,
they
proved
the
following
theorem.
*
Received
date:
2011-0
4-
02
Accepted
date:
2011-12-26
Foundation
item:
Supported
by
University National
Science
Foundation of Jiangsu Province
(09KJB110002).
Biography:
Zheng
Famei (1970-), male, born
at
Suqian, Jiangsu, lecture, master, major
in
prob-
ability theory and statistics and its applications.
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