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 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 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 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].

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. 

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 On probabilities of large deviations

1970 ◽  
Vol 3 (2) ◽  
pp. 277-285 ◽  
Author(s):  
Vijay K. Rohatgi
Keyword(s):  
Large Deviations ◽  
Rate Of Convergence ◽  
Random Variables ◽  
Probabilities Of Large Deviations ◽  
Independent Identically Distributed

Let {Xn} be a sequence of independent identically distributed random variables and let The rate of convergence of probabilities , where 2 > r > 1, is studied.

scholarly journals On the rates of convergence to symmetric stable laws for distributions of normalized geometric random sums

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3073-3084
Author(s):  
Tran Hung ◽  
Phan Kien
Keyword(s):  
Rate Of Convergence ◽  
Random Variables ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Stable Laws ◽  
Random Sums ◽  
Probability Metric ◽  
Geometric Random Variable ◽  
Independent Identically Distributed

Let X1,X2,... be a sequence of independent, identically distributed random variables. Let ?p be a geometric random variable with parameter p?(0,1), independent of all Xj, j ? 1: Assume that ? : N ? R+ is a positive normalized function such that ?(n) = o(1) when n ? +?. The paper deals with the rate of convergence for distributions of randomly normalized geometric random sums ?(?p) ??p,j=1 Xj to symmetric stable laws in term of Zolotarev?s probability metric.

scholarly journals Improving the Asmussen–Kroese-Type Simulation Estimators

2012 ◽  
Vol 49 (4) ◽  
pp. 1188-1193 ◽  
Author(s):  
Samim Ghamami ◽  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Nonnegative Integer ◽  
Rare Event ◽  
Random Variables ◽  
Random Variable ◽  
Heavy Tailed ◽  
Independent Identically Distributed

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

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