On the Asymptotic Normality of a Simple Batch Epidemic | ||||
| The Egyptian Statistical Journal | ||||
| Article 2, Volume 27, Issue 1, June 1983, Page 16-28 | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/esju.1983.316609 | ||||
 View on SCiNiTO | ||||
| Authors | ||||
| Abdul-Hadi N. Ahmed; Abdul-Latif Younis | ||||
| 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. | ||||
| Keywords | ||||
| Asymptotic Normality; Central Limit Theorem; Covariance Function; Gaussian Process; Simple Batch Epidemic; Stochastic Processes; Subsequences | ||||
|
Statistics Article View: 21 |
||||
