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

#### Abstract

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.

#### Keywords

#### Full Text:

PDF#### References

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