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

This observe issues the asymptotic habit of a Markov procedure bought from normalized items of self reliant and identically disbursed random matrices. The susceptible convergence of this approach is proved, in addition to the legislation of huge numbers and the imperative restrict theorem.

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K→∞ By (I), we know V lim sup Sn /n > μ = 0, thus n→∞ V lim sup Sn /n = μ n→∞ = V lim sup Sn /n = μ + V lim sup Sn /n > μ n→∞ ≥ V lim sup Sn /n ≥ μ n→∞ n→∞ = 1. 30 Z. Chen and P. Wu Considering the sequence of (−Xn )∞ n=1 , from E[−Xi ] ≡ −μ, we have V lim sup(−Sn )/n = −μ = 1. n→∞ Therefore, V lim inf Sn /n = μ = 1. n→∞ The proof of (II) is complete. Acknowledgements. This work is supported partly by the National Basic Research (973)Program of China (No. 2007CB814901) and WCU(World Class University) program of the Korea Science and Engineering Foundation (R31-20007).

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). 3 Pseudo-operations We have seen that the Lehrer-concave integral as also the Choquet integral are strongly related to the usual operations of addition + and multiplication · on the interval [0, ∞].

Xia 4. : BSDE and related g-expectation. , Mazliak, L. ) Backward Stochastic Diﬀerential Equations. Pitman Research Notes in Mathematics Series, vol. 364, pp. 141–159 (1997a) 5. : BSDE and stochastic optimizations. , Wu, L. ) Topics in Stochastic Analysis. Science Press, Beijing (1997b) (in Chinese) 6. : Risk aversion in the small and in the large. Econometrica 32, 122–136 (1964) Riesz Type Integral Representations for Comonotonically Additive Functionals Jun Kawabe Abstract. The Daniell-Stone type representation theorem of Greco leads us to another proof and an improvement of the Riesz type representation theorem of Sugeno, Narukawa, and Murofushi for comonotonically additive, monotone functionals.