scholarly journals Normal approximation for sum of random number of summands

2021 ◽  
Vol 47 ◽  
Author(s):  
Leonas Saulis ◽  
Dovilė Deltuvienė
Keyword(s):  
Large Deviations ◽  
Random Number ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Negative Integer

Normal aproximationof sum Zt =ΣNti=1Xi of i.i.d. random variables (r.v.) Xi , i = 1, 2, . . . with mean EXi = μ and variance DXi = σ2 > 0 is analyzed taking into consideration large deviations. Here Nt is non-negative integer-valued random variable, which depends on t , but not depends at Xi , i = 1, 2, . . ..

scholarly journals Theorems on large deviations for the sum of random number of summands

2010 ◽  
Vol 51 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Leonas Saulis
Keyword(s):  
Large Deviations ◽  
Rate Of Convergence ◽  
Random Number ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Compound Process ◽  
Independent Identically Distributed

In this paper, we present the rate of convergence of normal approximation and the theorem on large deviations for a compound process Zt = \sumNt i=1 t aiXi, where Z0 = 0 and ai > 0, of weighted independent identically distributed random variables Xi, i = 1, 2, . . . with  mean EXi = µ and variance DXi = σ2 > 0. It is assumed that Nt is a non-negative integervalued random variable, which depends on t > 0 and is independent of Xi, i = 1, 2, . . . .

scholarly journals The discount version of large deviations for a randomly indexed sum of random variables

2011 ◽  
Vol 52 ◽  
pp. 369-374
Author(s):  
Aurelija Kasparavičiūtė ◽  
Leonas Saulis
Keyword(s):  
Large Deviations ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Negative Integer ◽  
Exponential Inequalities ◽  
Sum Of Random Variables ◽  
Independent Identically Distributed

In this paper, we consider a compound random variable Z = \sum^N_{j=1} vjXj , where 0 < v < 1, Z = 0, if N = 0. It is assumed that independent identically distributed random variables X1,X2, . . . with mean EX = μ and variance DX =σ2 > 0 are independent of a non-negative integer-valued random variable N. It should be noted that, in this scheme of summation, we must consider two cases: μ\neq 0 and μ = 0. The paper is designatedto the research of the upper estimates of normal approximation to the sum ˜ Z = (Z −EZ)(DZ)−1/2, theorems on large deviations in the Cramer and power Linnik zones and exponential inequalities for P( ˜ Z > x).    

scholarly journals Discounted payments theorems for large deviations

2016 ◽  
Vol 57 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Dovilė Deltuvienė
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Negative Integer ◽  
Nonuniform Estimate ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Independent Identically Distributed

  Let Z(t) = Σ j=1N(t) Xj, t ≥ 0, be a stochastic process, where Xj are independent identically distributed random variables, and N(t) is non-negative integer-valued process with independent increments. Throughout, we assume that N(t) and Xj are independent. The paper considers normal approximation to the distribution of properly centered and normed random variable Zδ =∫0∞e- δt dZ(t), δ > 0, taking into consideration large deviations both in the Cramér zone and the power Linnik zones. Also, we obtain a nonuniform estimate in the Berry–Essen inequality. 

scholarly journals Large deviations for weighted random sums

2013 ◽  
Vol 18 (2) ◽  
pp. 129-142 ◽  
Author(s):  
Aurelija Kasparavičiūtė ◽  
Leonas Saulis
Keyword(s):  
Large Deviations ◽  
Normal Approximation ◽  
Random Variables ◽  
Main Idea ◽  
Random Variable ◽  
Tail Probability ◽  
Random Sums ◽  
Exponential Inequalities ◽  
The Difference ◽  
Independent Identically Distributed

In the present paper we consider weighted random sums ZN = ∑j=1NajXj, where 0 ≤ aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum ~ZN = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(~ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers [1,2].

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

scholarly journals On the probabilities of large deviations for sums of random number of independent random variables

1967 ◽  
Vol 7 (3) ◽  
pp. 513-516
Author(s):  
V. Statulevičius
Keyword(s):  
Large Deviations ◽  
Random Number ◽  
Random Variables ◽  
Independent Random Variables ◽  
Probabilities Of Large Deviations

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. А. Статулявичюс. О вероятностях больших уклонений для суммы случайного числа независимых случайных величин V. Statulevičius. Didelių atsilenkimų tikimybių nepriklausomų atsitiktinių dydžių sumai su atsitiktiniu dėmenų skaičiumi klausimu

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