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Cavazos-Cadena R., Hernandez-Hernandez D.'s A central limit theorem for normalized products of random PDF

By Cavazos-Cadena R., Hernandez-Hernandez D.

This notice matters the asymptotic habit of a Markov technique got from normalized items of self sufficient and identically dispensed random matrices. The susceptible convergence of this approach is proved, in addition to the legislation of enormous numbers and the critical restrict theorem.

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Sulla rappresentazione di funzionali mediante integrali. Rend. Sem. Mat. Univ. Padova 66, 21–42 (1982) 5. : Regular fuzzy measure and representation of comonotonically additive functional. Fuzzy Sets and Systems 112, 177–186 (2000) 6. : Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht (1995) 7. : Integral representation without additivity. Proc. Amer. Math. Soc. 97, 255–261 (1986) 8. : Choquet integral and fuzzy measures on locally compact space. Fuzzy Sets and Systems 99, 205–211 (1998) 9.

2-monotone or convex, if it satisfies the following inequality for all A, B ∈ 2N : v(A ∪ B) + v(A ∩ B) v(A) + v(B). The functional f → Cv (f ) is concave if and only if v is supermodular, see [12]. Lehrer concave integral is given in the following definition, [11]. Definition 1. Concave integral of a measurable function f : Ω → [0, ∞[ is given by (L) ai v(Ai ) | f dv = sup i∈I where Ai , i ∈ I, are measurable. ai 1 Ai i∈I f, I is finite , ai 0 , 46 R. Mesiar, J. Li, and E. Pap Note that the equality (L) 1A dv = v(A) is violated, in general, and it holds only for supermodular v (for all A ∈ A).

Then f ∧ b = f for a sufficiently large b > 0, so that limb→∞ I(f ∧ b) = I(f ). (3) Fix f ∈ F, and let gn := f − f ∧ (1/n) for each n ∈ N. Since gn ∼ f ∧(1/n), the comonotonic additivity of I implies I(f ) = I(gn )+I(f ∧(1/n)). Since f ∧ (1/n) ∞ ≤ 1/n for all n ∈ N, by the boundedness of I, I(gn ) → I(f ) as n → ∞. Take ε > 0 arbitrarily, and chose δ > 0 such that δ < 1/n0 and I(f ) − I(gn0 ) < ε. Let 0 < a < δ. Then, f − f ∧ a ≥ gn0 , so that I(f − f ∧ a) ≥ ✷ I(gn0 ) > I(f ) − ε. Thus, we have lima→+0 I(f − f ∧ a) = I(f ).

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A central limit theorem for normalized products of random matrices by Cavazos-Cadena R., Hernandez-Hernandez D.


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