A Density Version of A Local Limit Theorem for Properly Normalised Weighted Sums
Keywords:
Limit distribution, Domain of partial attraction, Moment convergence, Local limit theoremAbstract
Let n X ,n 1 be a sequence of i.i.d. random variables with a common distribution function F with
1 EX 0 . Let n
n k
k 1
S X , n 1
and
n
n k
k 1
k
T f X
n
, where f is positive, non -
decreasing and continuous on [0,1] with f(1) 1. Let k n be an integer sequence such
that,
1
n n n n
k k k k
Z B T A , where n
k
(A ) and n
k
(B 0) are sequences of norming
constants with n
k
B as k. We obtain a moment convergence result for n
k
Z and
a density version of a local limit theorem when distribution function F belongs to the domain
of partial attraction of a semistable law.
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