On a Local Limit Theorem for the Maximum of Sums of Independent Random Variables

1977 ◽  
Vol 21 (2) ◽  
pp. 384-385 ◽  
Author(s):  
S. V. Nagaev ◽  
M. S. Eppel’
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables

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 An improvement of convergence rate in the local limit theorem for integral-valued random variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tatpon Siripraparat ◽  
Kritsana Neammanee
Keyword(s):  
Limit Theorem ◽  
Convergence Rate ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Moment Condition ◽  
Formula One ◽  
Explicit Constant ◽  
Independent Integral

AbstractLet $X_{1}, X_{2}, \ldots , X_{n}$ X 1 , X 2 , … , X n be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ S n = ∑ j = 1 n X j . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ S n . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ O ( 1 n ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ C n . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ S n . Our constants are less complicated than before, and thus easy to use.

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.

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

A non-uniform rate of convergence in a local limit theorem

1978 ◽  
Vol 84 (2) ◽  
pp. 351-359 ◽  
Author(s):  
Sujit K. Basu
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Standard Normal ◽  
The Common ◽  
Necessary And Sufficient

AbstractLet {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

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