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

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 style approximation theorem which issues the matter of approximating a functionality f by way of a series {Lnf } of confident linear operators.

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Extra info for A -Statistical extension of the Korovkin type approximation theorem

Example text

K , define the transition matrix M as ⎛p ⎞ p ... p 1,1 ⎜ p2,1 M=⎜ ⎝ .. p K ,1 1,2 p2,2 .. p K ,2 ... . ··· 1, K p2, K ⎟ .. ⎟ ⎠. pK , K If we sum across the ith row of the matrix (i = 1, . . , K ), we exhaust the states we can go to from state i, and we must have K pi j = 1, for each row i = 1, . . , K . j=1 Of interest is the state probability row vector π (n) = π1(n) π2(n) ··· π K(n) , where the component πi(n) is the a priori probability of getting in state i after n transitions from the beginning moment of time.

Xik ), for any arbitrary permutation (i 1 , . . , i k ) of the indexes in {1, . . , k}. That is to say, P( X1 ≤ x1 , . . , Xk ≤ xk ) = P( Xi1 ≤ x1 , . . , Xik ≤ xk ), or equivalently, P( X1 ≤ x1 , . . , Xk ≤ xk ) = P( X1 ≤ xi1 , . . , Xk ≤ xik ), for any arbitrary permutation (i 1 , . . , i k ) of the indexes in {1, . . , k}. For events A1 , . . , Ak , we say they are exchangeable, when their indicators 1 A1 , . . , 1 Ak are exchangeable random variables (the indicator 1E of the event E is a random variable that assumes the value 1, when E occurs, and assumes the value 0 otherwise).

One could think of the variable index as the discrete time. One says that the system is in state i at time n, if Xn = i. These random variables are said to be a finite Markov chain, if the future state depends only on the current state. If we are given the full history since the beginning, the next state at time n + 1 is determined only by the state at time n. That is to say, the system is a (homogeneous finite-state) Markov chain if pi j := P( Xn+1 = j | X0 = x0 , X1 = x1 , . . , Xn−1 = xn−1 , Xn = i) = P( Xn+1 = j | Xn = i), which is simply the probability of making the transition to state j, if the system is in state i, and it depends only on the selection of the future state j and the present state i, regardless of how we got to it.

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