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316609

On the Asymptotic Normality of a Simple Batch Epidemic

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Last updated: 05 Jan 2025

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Abstract

In Section 2 we show that for some subsequences of the natural numbers {Lₐ(N)}, N = 1,2,...,n,..., α ∈ A, the sequences {S(Lₐ(N))} ,N = 1,2,...,n,..., α ∈ A, are asymptotically normal. The classical central limit theory does not apply in our case, since the summands (i.e., µ(N,Dⱼ₋₁) Uⱼ) are dependent and the number of summands (i.e., R [Lₐ(N)]) is random. We overcome those difficulties by using a normal central limit theorem due to A. Dvoretzky (1972). In Section 3 we prove that the sequence of stochastic processes given by {2λ/(√m sin(λmt)) [(Xₙ(t))/N - sin²(λmt/2)] √(N/lnN)} 0 < t < π/λm, N = 1,2,...,n, ...,                                             (1.7) converges in law to a centered Gaussian process with covariance function 1 for 0 < t ≤ s < π/λm when N →∞.  Proofs in Section 3 are based on equation (1.5) and the results of Section 2. Section 4 is devoted to the proofs of technical Lemmas needed in Sections 2 and 3.

DOI

10.21608/esju.1983.316609

Keywords

asymptotic normality, Central Limit Theorem, Covariance Function, Gaussian Process, Simple Batch Epidemic, Stochastic Processes, Subsequences

Authors

First Name

Abdul-Hadi

Last Name

Ahmed

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N.

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Orcid

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First Name

Abdul-Latif

Last Name

Younis

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Volume

27

Article Issue

1

Related Issue

43412

Issue Date

1983-06-01

Receive Date

2023-09-07

Publish Date

1983-06-01

Page Start

16

Page End

28

Print ISSN

0542-1748

Online ISSN

2786-0086

Link

https://esju.journals.ekb.eg/article_316609.html

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https://esju.journals.ekb.eg/service?article_code=316609

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2

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Original Article

Type Code

1,914

Publication Type

Journal

Publication Title

The Egyptian Statistical Journal

Publication Link

https://esju.journals.ekb.eg/

MainTitle

On the Asymptotic Normality of a Simple Batch Epidemic

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Article

Created At

28 Dec 2024