Ahmed, A., Younis, A. (1983). On the Asymptotic Normality of a Simple Batch Epidemic. The Egyptian Statistical Journal, 27(1), 16-28. doi: 10.21608/esju.1983.316609
Abdul-Hadi N. Ahmed; Abdul-Latif Younis. "On the Asymptotic Normality of a Simple Batch Epidemic". The Egyptian Statistical Journal, 27, 1, 1983, 16-28. doi: 10.21608/esju.1983.316609
Ahmed, A., Younis, A. (1983). 'On the Asymptotic Normality of a Simple Batch Epidemic', The Egyptian Statistical Journal, 27(1), pp. 16-28. doi: 10.21608/esju.1983.316609
Ahmed, A., Younis, A. On the Asymptotic Normality of a Simple Batch Epidemic. The Egyptian Statistical Journal, 1983; 27(1): 16-28. doi: 10.21608/esju.1983.316609
On the Asymptotic Normality of a Simple Batch Epidemic
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.
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