A local limit theorem for densities and large deviations of sums of a random number of random variables

1989 ◽  
Vol 47 (5) ◽  
pp. 2738-2742
Author(s):  
I. A. Melamed
Keyword(s):  
Limit Theorem ◽  
Large Deviations ◽  
Random Number ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit

Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (04) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

A local limit theorem for attractions under a stable law

1980 ◽  
Vol 87 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Sujit K. Basu ◽  
Makoto Maejima
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Stable Law ◽  
Absolutely Continuous ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.

scholarly journals A note on a local limit theorem for Wiener space valued random variables

Bernoulli ◽  
2016 ◽  
Vol 22 (4) ◽  
pp. 2101-2112 ◽  
Author(s):  
Alberto Lanconelli ◽  
Aurel I. Stan
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Wiener Space
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