By Erkus E., Duman O.

During this paper, utilizing the idea that ofA-statistical convergence that is a regular(non-matrix) summability strategy, we receive a normal Korovkin variety approximation theorem which matters the matter of approximating a functionality f by way of a series {Lnf } of optimistic linear operators.

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Then ∞ WH (t) = n=1 sin(xn t) Xn + xn ∞ n=1 1 − cos(yn t) Yn . 114) in the sense that random processes on both sides have the same ﬁnite-dimensional distributions. The random variables Xi , i ≥ 1, and Yi , i ≥ 1, are independent Gaussian random variables with mean zero and with the variances given by 2 −2H −2 xn J1−H (xn ), V ar(Xn ) = 2CH 2 −2H −2 V ar(Yn ) = 2CH yn J−H (yn ), and 2 = CH 1 (1 + 2H ) sin(πH ). s. 114) generalizes the result on Karhunen–Loeve-type expansion for Brownian motion. 114) can be used for simulating an fBm sample from Gaussian samples.

Strong consistency Let LT (θ ) denote the Radon–Nikodym derivative dPθT /dP0T . The maximum likelihood estimator (MLE) θˆT is deﬁned by the relation LT (θˆT ) = sup LT (θ ). 11) θ∈ We assume that there exists a measurable maximum likelihood estimator. Sufﬁcient conditions can be given for the existence of such an estimator (cf. 2 in Prakasa Rao (1987)). Note that QH,θ (t) = d dwtH t kH (t, s) 0 C(θ, s) ds σ (s) t d a(s, X(s)) d kH (t, s) ds + θ H σ (s) dwt 0 dwtH = J1 (t) + θ J2 (t) (say). 13) and the likelihood equation is given by T T J2 (t)dZt − 0 0 (J1 (t) + θ J2 (t))J2 (t)dwtH = 0.

18) Note that the quadratic variation Z of the process Z is the same as the quadratic variation M H of the martingale M H which in turn is equal to wH . 73) in Chapter 1. s. [Pθ0 ] i+1 i where (n) =T 0 = t0(n) < t1(n) < · · · < tn−1 is a partition of the interval [0, T ] such that (n) − ti(n) | → 0 sup |ti+1 0≤i≤n−1 as n → ∞. ) is an unknown constant σ , the above property can be used to obtain a strongly consistent estimator of σ 2 based on the continuous 50 STATISTICAL INFERENCE FOR FRACTIONAL DIFFUSION PROCESSES observation of the process X over the interval [0, T ].

### A -Statistical extension of the Korovkin type approximation theorem by Erkus E., Duman O.

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