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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Additional info for A central limit theorem for normalized products of random matrices
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 . Lehrer concave integral is given in the following deﬁnition, . 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 ).
A central limit theorem for normalized products of random matrices by Cavazos-Cadena R., Hernandez-Hernandez D.