almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

symmetrization approach to concentration inequalities for empirical processes we introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional as an application we give a new easy proof of talagrand s concentration inequality for empirical processes where besides symmetrization we use only talagrand s concentration inequality on the discrete cube 0 1 n as another application of this technique we prove new vapnik chervonenkis type inequalities for example for vc classes of functions we prove a classical inequality of vapnik and chervonenkis only with normalization by the sum of variance and sample variance

a recursive deconvolution approach to disturbance reduction active noise control anc uses an estimate of the noise affecting a system in order to remove its effect from the output in some applications it is possible to directly measure the noise and a feedforward compensator can be used other applications require instead that the noise be estimated from its effect on the system this results in an adaptive feedback anc see a previous paper by gan and kuo the identification of the disturbance from the output of the linear system is a deconvolution problem in this paper we study a deconvolution technique for the active reduction of the influence of the disturbance on the output of a linear control system the regulator that we present does not affect the behavior of the system in the absence of disturbances

precise asymptotics in the self normalized law of the iterated logarithm abstract let x x 1 x 2 be i i d nondegenerate random variables with zero means s n j 1 n x j and v n 2 j 1 n x j 2 we investigate the precise asymptotics in the law of the iterated logarithm for self normalized sums s n v n also for the maximum of self normalized sums max 1 k n s k v n when x belongs to the domain of attraction of the normal law

concentration inequalities large and moderate deviations for self normalized empirical processes we consider the supremum w n of self normalized empirical processes indexed by unbounded classes of functions f such variables are of interest in various statistical applications for example the likelihood ratio tests of contamination using the herbst method we prove an exponential concentration inequality for w n under a second moment assumption on the envelope function of f this inequality is applied to obtain moderate deviations for w n we also provide large deviations results for some unbounded parametric classes f

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

cramer type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series where 1 j q and q n 1 2 the periodogram is a fundamental tool in spectral analysis and is often used to detect periodic patterns in various real applications such as the analysis of gene expression data and the study of earthquake theoretical properties of the periodogram have been extensively studied an chen and hannan 1983 obtained the logarithm law for the maximum of the periodogram davis and mikosch 1999 mikosch resnick and samorodnitsky 2000 lin and liu 2009a obtained the asymptotic distribution for the maximum of the periodogram under the i i d and linear process cases the heavy tailed case and nonlinear time series case respectively fay and soulier 2001 proved the central limit theorem for functionals of the periodogram shao and wu 2007 obtained asymptotic distributions for the periodogram and the empirical distribution function of the periodogram for a wide class of nonlinear processes when xn are independent and identically distributed i i d random variables with var x1 2 and e x1 2 0 davis and mikosch 1999 show that

algebraic characterization of controllability subspaces in the current base and its practical computation from the concepts of input chain and controllability chain introduced by heymann and in view of the generalized eigenvector assignment a new algebraic characterization of controllability subspaces css is developed compared with the previous ones based on matrix pencil theory it presents both the advantages of being stated in the current base and of being derived without resorting to use of canonical forms a practical algebraic algorithm is provided in order to compute the controllable structure of a given pair a b this approach is then applied to characterize css contained in a given subspace and span css via eigenvector assignment

towards a universal self normalized moderate deviation this paper is an attempt to establish a universal moderate deviation for self normalized sums of independent and identically distributed random variables without any moment condition the exponent term in the moderate deviation is specified when the distribution is in the centered feller class an application to the law of the iterated logarithm is given

darling erdős theorem for self normalized sums let x x1 x2 be i i d nondegenerate random variables sn nj 1xj and v2n nj 1x2j we investigate the asymptotic vspace 1pt behavior in distribution of the maximum of self normalized sums max1 k nsk vk and the law of the iterated logarithm for self normalized sums sn vn when x belongs to the domain of attraction of the normal law in this context we establish a darling erd s type theorem as well as an erd s feller kolmogorov petrovski type test for self normalized sums

performance of the bayesian online algorithm for the perceptron in this letter we derive continuum equations for the generalization error of the bayesian online algorithm bona for the one layer perceptron with a spherical covariance matrix using the rosenblatt potential and show by numerical calculations that the asymptotic performance of the algorithm is the same as the one for the optimal algorithm found by means of variational methods with the added advantage that the bona does not use any inaccessible information during learning

an almost sure central limit theorem for self normalized partial sums let x i i 1 be a sequence of i i d random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance denote s email protected i 1 nx i v n email protected i 1 nx i 2 then an almost sure central limit theorem for self normalized partial sums s n v n is studied under a mild condition in this paper

an almost sure central limit theorem for self normalized products of sums of i i d random variables abstract let x x 1 x 2 be a sequence of independent and identically distributed positive random variables with ex 0 in this paper we show that the almost sure central limit theorem for self normalized products of sums holds only under the assumptions that x belongs to the domain of attraction of the normal law

matrix pencil characterization of almost a z invariant subspaces a classification of geometric concepts the equivalence between the algebraic matrix pencil characterization of the sub spaces of the extended geometric theory and their dynamic characterization 13 established as a result a complete classification of almost a b invariant a b invariant almost controllability and controllability subspaces is derived in terms of matrix pencil invariants the frequency propagation aspects of infinite spectrum a b invariant subspaces are investigated and it is shown that they are limits of closed loop eigenspaces with arbitrarily largo eigenvalues finally the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed loop eigenspaces and eigenvalues under scalar gain output feedback is discussed

an exponential nonuniform berry esseen bound for self normalized sums in this paper we shall derive exponential nonuniformberry esseen bounds in the central limit theorem for self normalized sums we show that the size of the error can be reduced considerably by replacing the usual standardization by self normalization in particular we establish the exponential bounds for the probability of the self normalized sums under the condition that the third moment is finite whereas an exponential moment assumption is required for the standardized sums applications to t statistics and the probabilities of moderate deviations of self normalized sums are also discussed

exact inequalities for sums of asymmetric random variables with applications let rm bs 1 dots rm bs n be independent identically distributed random variables each having the standardized bernoulli distribution with parameter p in 0 1 let m p 1 p 2 p 2 2 sqrt p p 2 4 p 2 if 0 p le frac12 and m p 1 if frac12 le p 1 let m ge m p let f be such a function that f and f are nondecreasing and convex then it is proved that for all nonnegative numbers c 1 dots c n one has the inequality mathsf e f c 1 mathrm b s 1 dots c n mathrm b s n le mathsf e f big s m mathrm b s 1 dots mathrm bs n big where s m big frac1n sum i 1 n c i 2m big frac1 2m the lower bound m p on m is exact for each p in 0 1 moreover operatorname mathsf e f c 1 mathrm b s 1 dots c n mathrm b s n is schur concave in c 1 2m ldots c n 2m

deviation inequalities for martingales with applications using changes of probability measure developed by grama and haeusler 2000 18 we extend the deviation inequalities of lanzinger and stadtmuller 2000 26 and fuk and nagaev 1971 15 to the case of martingales our inequalities recover the best possible decaying rate in the independent case in particular these inequalities improve the results of lesigne and voln 2001 27 under a stronger condition that the martingale differences have bounded conditional moments applications to linear regressions with martingale difference innovations weak invariance principles for martingales and self normalized deviations are provided in particular we establish a type of self normalized deviation bounds for parameter estimation of linear regressions such type bounds have the advantage that they do not depend on the distribution of the regression random variables

precise asymptotics in complete moment convergence for self normalized sums let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and v n 2 x 1 2 x n 2 ex 2i x x is a slowly varying function at we prove that for any b 2 d 2 b 1 email protected email protected b d 1 email protected n 2 logn d 2 bne s n v n 2i s n email protected n logn 1 b email protected n b d 1 b d 1 2 where n is a standard normal random variable

precise asymptotics in the deviation probability series of self normalized sums abstract let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and w n 2 x 1 2 x n 2 in the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self normalized sums s n w n for positive functions g x x x and x we obtain the precise asymptotics for the following deviation probabilities of self normalized sums n 1 g n n e s n w n 2 i s n w n n n the results given can be considered as the generalization of that in the complete moment convergence law of iterated logarithm and large deviation for self normalized sums

large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation let x n 1 leq n infty be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index 0 alpha 2 the limit of x n 1 log p s n max x i geq x n is found when x n to infty and x n n to 0 the large deviation result is used to prove the law of the iterated logarithm for the self normalized partial sums

moderate deviations for finite population student s statistic in this paper moderate deviations for finite population are given under weaker conditions as by products some exponential inequalities for finite population are also established

a normal comparison inequality and its applications let xi xi i 1 leq i leq n and eta eta i 1 leq i leq n be standard normal random variables with covariance matrices r 1 r ij 1 and r 0 r ij 0 respectively slepian s lemma says that if r ij 1 geq r ij 0 for 1 leq i j leq n the lower bound p xi i leq u mb for 1 leq i leq n p eta i leq u mb for 1 leq i leq n is at least 1 in this paper an upper bound is given the usefulness of the upper bound is justified with three concrete applications i the new law of the iterated logarithm of erd s and revesz ii the probability that a random polynomial does not have a real zero and iii the random pursuit problem for fractional brownian particles in particular a conjecture of kesten 1992 on the random pursuit problem for brownian particles is confirmed which leads to estimates of principal eigenvalues

on line learning in parity machines the optimal algorithm for on line learning in the tree k parity machine is studied we introduce a set of recursion relations for the relevant probability distributions which permit study of the general k case the generalization error curve is determined and shown to decay to zero for large as even in the presence of noise there is no critical noise level the dynamics of on line learning is studied analytically near the origin in the absence of previous knowledge the learning dynamics has a fixed point at previous knowledge is needed in at least k 1 branches for the learning to take place

asymptotic distribution of the largest off diagonal entry of correlation matrices suppose that we have observations from a dimensional population we are interested in testing that the variates of the population are independent under the situation where goes to infinity as a test statistic is chosen to be where is the sample correlation coefficient between the th coordinate and the th coordinate of the population under an independent hypothesis we prove that the asymptotic distribution of is an extreme distribution of type by using the chen stein poisson approximation method and the moderate deviations for sample correlation coefficients as a statistically more relevant result a limit distribution for where is spearman s rank correlation coefficient between the th coordinate and the th coordinate of the population is derived

learning strategy for the binary perceptron pursuing the work of penney and sherrington we determine the optimal continuous weight perceptron which on clipping correctly predicts the largest number of weights for the binary perceptron with maximum stability we calculate the fraction of correctly predicted binary weights when only the continuous weights stronger than a certain threshold are clipped we finally carry out simulations for a perceptron with 50 weights to test the practicability of different learning strategies

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

generalization from correlated sets of patterns in the perceptron generalization is a central aspect of learning theory here we proposea 13 a framework that explores an auxiliary task dependent notion of gener a 13 alization and attempts to quantitatively answer the following question a 13 given two sets of patterns with a given degree of dissimilarity how easilya 13 will a network be able to unify their interpretation this is quantifieda 13 by the volume of the configurations of synaptic weights that classify thea 13 two sets in a similar manner to show the applicability of our idea ina 13 a concrete setting we compute this quantity for the perceptron a sim a 13 ple binary classifier using the classical statistical physics approach in thea 13 replica symmetric ansatz in this case we show how an analytical ex a 13 pression measures the distance based capacity the maximum load ofa 13 patterns sustainable by the network at fixed dissimilarity between pat a 13 terns and fixed allowed number of errors this curve indicates that gen a 13 eralization is possible at any distance but with decreasing capacity wea 13 propose that a distance based definition of generalization may be usefula 13 in numerical experiments with real world neural networks and to explorea 13 computationally sub dominant sets of synaptic solutions

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

a kind of complete moment convergence for self normalized sums let x x n n 1 be a sequence of independent and identically distributed i i d random variables with x in the domain of attraction of the normal law and ex 0 in this note we obtain a new kind of complete moment convergence for self normalized sums

bayes optimal performance in a discrete space we study a simple model of unsupervised learning where the single symmetry breaking vector has binary components 1 we calculate exactly the bayes optimal performance of an estimator which is required to lie in the same discrete space we also show that except for very special cases such an estimator cannot be obtained by minimization of a class of variationally optimal potentials

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

bayesian learning versus optimal learning abstract we consider the optimal performance that may be reached in the problem of learning the symmetry breaking direction of a cloud of p n points in a n dimensional space the performance is measured through the overlap ropt between the true symmetry breaking direction and the learnt one depending on the problem the learning curves ropt may present discontinuities we show that close to these bayesian learning is not optimal

new decoupling invariants the essential orders the aim of this paper is to introduce a new list of integers which is invariant under the action of biproper compensation and which plays a key role in the solution of the row by row decoupling problem several equivalent characterizations are provided within both the geometric and the transfer matrix approaches

strassen type law of the iterated logarithm for self normalized increments of sums abstract let x x n n 1 be a sequence of independent identically distributed random variables with e x 0 and assume that e x 2 i x x is slowly varying as x i e x is in the domain of attraction of the normal law in this paper it is shown that a strassen type functional law of the iterated logarithm holds for self normalized increments of sums of such random variables

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance