By Han-fu Chen, Lei Guo

Identifying the input-output dating of a approach or researching the evolutionary legislations of a sign at the foundation of statement info, and utilising the developed mathematical version to predicting, controlling or extracting different worthwhile info represent an issue that has been drawing loads of consciousness from engineering and gaining an increasing number of value in econo­ metrics, biology, environmental technology and different comparable components. over the past 30-odd years, examine in this challenge has swiftly constructed in quite a few parts lower than diversified phrases, akin to time sequence research, sign processing and procedure identity. because the randomness often exists in actual platforms and in commentary info, and because the random method is usually used to version the uncertainty in platforms, it really is average to contemplate the article as a stochastic approach. In a few purposes id may be conducted off line, yet in different situations this can be most unlikely, for instance, whilst the constitution or the parameter of the method is dependent upon the pattern, or while the method is time-varying. In those circumstances we need to determine the procedure on-line and to regulate the keep watch over in keeping with the version that's presupposed to be impending the real method through the technique of identity. it is because there was an expanding curiosity in identity and adaptive regulate for stochastic platforms from either theorists and practitioners.

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We)). 4. g. [St], [HH]). Moreover these results, due to various constraints, are difficult to be used directly in the analysis of stochastic adaptive algorithms. In particular, in the order estimation problems of Chapter 5, when upper bounds for system orders are not available, a crucial step in analyzing estimation algorithms is to study asymptotic behaviors of sequences of the form max max 1

O. 6) {e > ~*} = U{e > b > a > ~*} a b > a > ~*) > O. 7) Let f3N(a, b) be the number of up-crossing of the interval (a, b) by {~n,Fn}, n ~ N. 1. 8) we have sup Ect + lal Ef3oo(a, b) = lim Ef3N(a, b) ~ N b N-+oo a - < 00. 7) means P(f3oo (a, b) = 00) > O. This contradicts Ef3oo(a, b) < 00. Hence P(C = c*) = 1, or Cn converges to a limit Cwhich may be infinite for some w. 2 we have Ee+ = Eliminfe: ~ liminfEe: ~ sup Ee: < 00 n-+oo and Ee- n-+oo = Eliminfc; n-+oo n ~ liminfEc,~ ~ n-+oo sup Ec;; n = sup(Ect - ECn) ~ sup(Ect - E6) < n since ECl ~ n ECn, n ::::: 1 for submartingales.

S. 22) limsup(2s n loglogsn) -;' n-+oo Xi i=l n ;=1 where n s~ = 2:E(x;IFi-d. s. I>< . 1-<6("". '-')=1 ) for any 6 > 0 [HHl. For analyzing stochastic adaptive control systems we need to know the behavior of partial sums of a martingale difference sequence with weights. 22) are crucial for us. s. and Mn is an Fn-measurable random matrix in the case where neither the boundedness of IIMiXi+111 nor the finiteness of Ell Mi Xi +1 II takes place. 3. Martingale Difference Sequence 41 In the sequel for a sequence of matrices {Md and a nondecreasing sequence of positive real numbers {bd, by Mn = O(b n ) we mean lim sup k--+oo lim k-+oo In particular, by such that Mk = IIMkll/bk < 00 IIMkll/bk = o.

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