Asymptotic properties of the method of observed mean for homogeneous random fields
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ASYMPTOTIC PROPERTIES OF THE METHOD OF OBSERVED MEAN FOR HOMOGENEOUS RANDOM FIELDS D. A. Gololobova and E. J. Kasitskaya b
UDC 519.21
Abstract. The empirical estimate of the unknown parameter of a homogenous (in restricted sense) random field with continuous time and continuous states observed in a circle is considered. The strong consistency of the estimate is proved. The conditions under which the estimates weakly converge to the Gaussian distribution are established. Keywords: method of observed mean, random field, probability, function, minimization, continuity. INTRODUCTION The method of observed mean [1–4] is one of the most efficient approximation methods in stochastic optimization theory. This is mainly due to its close relation with the statistical estimation and optimal control theory. The well developed theory of asymptotic estimation provides tools for the analysis of convergence and of the rate of convergence of iterative procedures of stochastic programming; moreover, stochastic programming methods allow approximate estimation of unknown parameters of stochastic systems. In the paper, we will consider optimization problems for stochastic systems based on observations of random fields in a bounded domain. PROBLEM STATEMENT AND AUXILIARY INFORMATION r r r Let {x ( t ) = x ( t , w ), t Î R m }, m ³ 1, be a homogeneous (in the restricted sense) random field on a complete probability r space (W , G , P ) with values in a metric space (Y , B (Y )) . Realizations x( t ) are assumed to be continuous on R m with probability one. A continuous non-negative function b : J ´ Y ® R is specified, where J is a closed subset of R l , l ³ 1. r r There are observations {x ( t ): || t ||< T }, T > 0 . It is required to find points of minimum and the minimum value of function r r r r (1) F ( u ) = M f ( u , x( 0) ), u Î J . Let us formulate some necessary r r auxiliary results. THEOREM 1 [5]. Let x( t ), t Î R , be a homogeneous (in the wide sense) random field with values in R; r r r r r r M( x )t ) = 0, and b( t ) = Mx ( t + s )x ( s ) . Realizations x( t ) are assumed to be Lebesgue measurable in R k with probability one. Denote r r r r r X ( t ) = ò x ( t )dt , B ( t ) = ò b( t )dt , L( t ) = ò dt . r ||t ||< t
r ||t ||< t
r ||t ||< t
Assume that 1
ò Then with probability one we have
0
r ö (ln r ) 2 æç 1 | B ( t )| dt ÷dr < ¥. ò k 2 ç t ÷ r è0 ø
X (t ) ® 0, t ® ¥ . L( t )
a
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. bV. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2013, pp. 160–167. Original article submitted January 10, 2013. 1060-0396/13/4903-0465
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2013 Springer Science+Business Media New York
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THEOREM 2 [6]. Let (W , U , P ) be a probability space and K be a compact subset of a Banach spaces with norm || · || . Assume that {QT ( s ) = QT ( s, w ) Î K ´ W , T Î R , T > 0} is a family of real functions satisfying the
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