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A Density Version of A Local Limit Theorem for Properly Normalised Weighted Sums

KN Raviprakash, K. Vidyalaxmi, Gooty Divanji


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 k
k 1
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
n n n n
k k k k
Z B T A    , where n
(A ) and n
(B  0) are sequences of norming
constants with n
B  as k. We obtain a moment convergence result for n
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.


Limit distribution, Domain of partial attraction, Moment convergence, Local limit theorem

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