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

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

This notice matters the asymptotic habit of a Markov approach received from normalized items of autonomous and identically disbursed random matrices. The vulnerable convergence of this technique is proved, in addition to the legislations of enormous numbers and the important restrict theorem.

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Let u B  0, v t   B f ( x )t B 1 , and u Nt   N f ( x )t   B f ( x )t B 1 N . Simple algebra shows that (u B , u N , v ) satisfies the above system (solve for v from the first equation and substitute it in the second equation). Therefore, x is a KKT point whenever d  0 (and is optimal if, for example, f is pseudoconvex). 33 In the first problem, the KKT system is given by: c  Hx  At u  0 Ax  y  b (1) (2) ut y  0 x  0, y  0, u  0. Since the matrix H is invertible, Equation (1) yields H 1c  x  H 1 At u  0.

K d    k d xk  x (1)  cos( k ), where  k is the angle between ( xk  x ) and d. Since d  T , we have that  k  0 and so cos( k )  1 and thus yk k  0 from (1). Consequently, we can define  :R  R n such that  k  (  k )  yk so that xk  x   k d  46  k  (  k )  S , k , with  (  k )  yk /  k  0 as  k  0 . Hence, d  W. Next, we show that if d  W , then d  T . For this purpose, let us note that if d  W , then the sequence {xk }  S , where xk  x   k d   k  (  k ), converges to x , and moreover, the  1  sequence  ( xk  x )  d  converges to the zero vector.

There exists xˆ  S with f ( xˆ )  f ( x ). Consider d  ( xˆ  x ). Then d  D since S is convex. Moreover, f ( x   d )  f ( xˆ  (1   ) x )   f ( xˆ )  (1   ) f ( x )  f ( x ), 0    1. 51, a contradiction. 3 similarly deals with nondifferentiable convex functions. If S  R n , then x is optimal  f ( x )t d  0, d  R n  f ( x )  0 (else, pick d  f ( x ) to get a contradiction). 56 Let x1 , x2  R n . Without loss of generality assume that h( x1 )  h( x2 ). Since the function g is nondecreasing, the foregoing assumption implies that g[h( x1 )]  g[h( x2 )], or equivalently, that f ( x1 )  f ( x2 ).

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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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