Document Type : Original Article
Authors
1
Institute of Statistical Studies & Research, Cairo University, Egypt
2
Department of Mathematics, Faculty of Science, Cairo University, Egypt
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
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