random coin tossing a sequence of heads and tails is produced by repeatedly selecting a coin from two possible coins and tossing it the second coin is tossed at renewal times in a renewal process and the first coin is tossed at all other times the first coin is fair prob heads 1 2 and the second coin is known either to be fair or to have known bias 0 1 prob heads letting u k prob there is a renewal at time k we show that if k 0 u k 2 we can determine using only the sequence of heads and tails produced if the second coin had bias or 0 if we show that this is not possible

t t 1 transformation is not loosely bernoulli the t t problem open since 1971 is to demonstrate that the t t transformation a naturally arising k transformation is not bernoulli it is shown here not even to be loosely bernoulli

ergodic properties of color records abstract in a series of papers culminating in the dissertation of the second author under the direction of p w kasteleyn several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black white colored lattice and this under the sole assumption that the coloring is stationary ergodic and independent of the walking in this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk k 0 of successive times needed for the walker to go from the kth to the k 1 st black point as an example of the type of results obtained we show that if a certain induced color record process is strongly mixing in the ergodic theoretic sense then the average nk tends to the inverse of the density of black points as k tends to infinity and similarly that the distribution of nk tends to a limit that can be explicitly calculated in terms of the distribution of n0 for a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing our main goal however is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records

brief paper output feedback variable structure control for linear systems with uncertainties and disturbances this paper proposes a dynamic output feedback variable structure controller for linear mimo systems with mismatched and matched norm bounded uncertainties and matched nonlinear disturbances the proposed controller consists of nonlinear and linear parts similar to the unit vector type of controller the nonlinear part of the new controller takes care only of matched uncertainties and disturbances and on the other hand the linear part with full dynamics completely handles mismatched uncertainties designing such a linear part leads to use a lyapunov function associated with the full states which achieves the global stability against the mismatched uncertainties the resulting criteria are furthermore converted into solvable ones by using the so called cone complementary linearization algorithm for bi convex problems

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

l cohomology is intersection cohomology abstract let x be a subanalytic compact pseudomanifold we show a de rham theorem for l forms on the nonsingular part of x we prove that their cohomology is isomorphic to the intersection cohomology of x in the maximal perversity

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

mean field criticality for percolation on planar non amenable graphs the critical exponents and are proved to exist and to take their mean field values for independent percolation on the following classes of infinite locally finite connected transitive graphs 1 non amenable planar with one end 2 unimodular with infinitely many ends

variations on a theme by mark kac mark kac s theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations most of which are formulated in terms of 0 1 processes whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0 this condition is dropped in part of the variations two others refer to the variance of the recurrence time and two to the poincare cycle of a dynamical system most variations consist in inequalities and formal identities for the mean first arrival time and subsequent recurrence times for the given state

sliding mode output feedback controller design using linear matrix inequalities considers a sliding mode controller which requires only output information and is static in nature the novelty of the approach is in the rationale and method used to synthesize the linear control component which involves a linear matrix inequality optimization the so called reachability condition used in sliding mode controller analysis is not required to be satisfied globally instead sliding is only expected to take place within a subset of the state space referred to as the sliding patch

output feedback multivariable global variable gain super twisting algorithm a novel output feedback super twisting algorithm design is proposed for uncertain multivariable plants in order to allow global finite time exact tracking in the presence of a rather general class of matched nonlinear disturbances which may depend on measured and or unmeasured states the algorithm is designed with variable gains this represents a significant extension of a recently proposed nondecoupled multivariable super twisting algorithm based on state feedback with fixed gains to the case of output feedback with variable gains

winning quick and dirty the greedy random walk as a strategy to complete games quickly we investigate one dimensional random walks where the step length increases deterministically upon each return to the origin when the step length after the kth return equals k the displacement of the walk x grows linearly in time asymptotically the probability distribution of displacements is a purely exponentially decaying function of x t the probability e t l for the walk to escape a bounded domain of size l at time t decays algebraically in the long time limit e t l l t2 consequently the mean escape time t lln l while tn l2n 1 for n 1 corresponding results are derived when the step length after the kth return scales as k for 0

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

dynamic output integral sliding mode control with disturbance attenuation this technical note proposes a dynamic output feedback sliding mode control algorithm for linear mimo systems with mismatched norm bounded uncertainties along with disturbances and matched nonlinear perturbations a control law is first designed to ensure that the system behavior can satisfy the reaching and sliding condition a scheme designed to combine the output dependent integral sliding surface with a full order compensator is then proposed through utilizing h infin control analytical technique once the system is in the sliding mode the proposed algorithm can guarantee robust stabilization and sustain the nature of performing disturbance attenuation when the solutions to two algebraic riccati inequalities can be found finally the feasibility of our proposed algorithm is illustrated using a numerical example

large deviations estimates for self intersection local times for simple random walk in mathbb z 3 we obtain large deviations estimates for the self intersection local times for a simple random walk in dimension 3 also we show that the main contribution to making the self intersection large in a time period of length n comes from sites visited less than some power of log n this is opposite to the situation in dimensions larger or equal to 5 finally we present an application of our estimates to moderate deviations for random walk in random sceneries

tail triviality for sums of stationary random variables we study tail sigma fields and loss of memory associated with sums of stationary integer valued random variables an application concerns convergence in distribution of interarrival times in zero one sequences

robust disturbance attenuation with stability for linear systems with norm bounded nonlinear uncertainties in this paper the problem of robust stabilization and robust disturbance attenuation is investigated for systems with linear nominal parts and norm bounded nonlinear uncertainties on both states and control inputs it is shown that the type of nonlinear uncertainty sets considered in this paper has an equivalent representation by linear uncertainty sets based on this key result and some standard riccati inequality approaches for robust control of linear uncertain systems a constructive design procedure is developed

mixing properties for random walk in random scenery consider the lattice z d d geq 1 together with a stochastic black white coloring of its points and on it a random walk that is independent of the coloring a local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position under weak assumptions on the probability distributions governing walk and coloring we prove asymptotic independence of local sceneries perceived at times 0 and n in the limit as n rightarrow infty and at times 0 and t k in the limit as k rightarrow infty where t k is the random k th hitting time of a black point an immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits i e of t k 1 t k as k rightarrow infty the limit distribution is expressed in terms of the distribution of the first hitting time t 1 the proof uses coupling arguments and ergodic theory

nonlinear integral type sliding surface for both matched and unmatched uncertain systems this note presents a new nonlinear integral type sliding surface which incorporates a virtual nonlinear nominal control to achieve prescribed specifications first the plant with matched uncertainties is considered the resultant closed loop system during ideal sliding mode behaves exactly like the nominal plant under the nonlinear nominal control which completely nullifies the matched uncertainties and consequently satisfies the prescribed specifications second the stability analysis of the proposed sliding mode for the systems with unmatched uncertainties is performed to exploit the stability conditions numerical results demonstrate the validity of the proposed concept

reconstructing a random scenery observed with random errors along a random walk path we show that an i i d uniformly colored scenery on observed along a random walk path with bounded jumps can still be reconstructed if there are some errors in the observations we assume the random walk is recurrent and can reach every point with positive probability at time k the random walker observes the color at her present location with probability 1 and an error y k with probability the errors y k k 0 are assumed to be stationary and ergodic and independent of scenery and random walk if the number of colors is strictly larger than the number of possible jumps for the random walk and is sufficiently small then almost all sceneries can be almost surely reconstructed up to translations and reflections

brief variable structure output feedback control design for a class of uncertain dynamic systems in this paper we consider the problem of designing a variable structure output feedback control vsofc law for a class of uncertain systems with mismatched uncertainty in the state matrix we derive existence conditions of linear sliding surfaces in terms of constrained linear matrix inequalities lmis and we give vsofc strategies we also give a simple design procedure and parameterizations of the linear switching surfaces together with solvability conditions for the given constrained lmis finally we give a design example in order to show the effectiveness of our method

on the number of infinite geodesics and ground states in disordered systems we study first passage percolation models and their higher dimensional analogs models of surfaces with random weights we prove that under very general conditions the number of lines or in the second case hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to as corollaries we show that in any dimensiond 2 the number of ground states of an ising ferromagnet with random coupling constants equals with probability one 2 or proofs employ simple large deviation estimates and ergodic arguments

stable limit laws for randomly biased walks on supercritical trees we consider a random walk on a supercritical galton watson tree with leaves where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree the biases are chosen independently on distinct edges each one according to a given law that satisfies a logarithmic nonlattice condition we determine the condition under which the walk is sub ballistic and in the sub ballistic regime we find a formula for the exponent 0 1 such that the distance x n moved by the walk in time n is of the order of n we prove a stable limiting law for walker distance at late time proving that the rescaled walk n x n converges in distribution to an explicitly identified function of the stable law of index this paper is a counterpart to ben arous et al ann probab 40 2012 280 338 in which it is proved that in the model where the biases on edges are taken to be a given constant there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement it is randomization of edge biases that is responsible for the emergence of the stable limit in the present article while also introducing further correlations into the model in comparison with the constant bias case the derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone the paper may be considered as a sequel to ben arous and hammond comm pure appl math 65 2012 1481 1527 since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical galton watson tree

cohomology of flat currents on definable pseudomanifolds abstract we show that the cohomology of flat currents on definable pseudomanifolds in polynomially bounded o minimal structures is isomorphic to intersection cohomology in the top perversity

retrieving random media benjamini asked whether the scenery reconstruction methods of matzinger see e g 21 22 20 can be done in polynomial time in this article we give the following answer for a 2 color scenery and simple random walk with holding we prove that a piece of the scenery of length of the order 3 n around the origin can be reconstructed up to a reflection and a small translation with high probability from the first 2 310 n observations with a constant 0 independent of n thus the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct the probability that the reconstruction fails tends to 0 as n

growth of uniform infinite causal triangulations we introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added a relation to a random walk on the integer half line is shown this relation is used to estimate the geodesic distance of a given triangle to the rooted boundary in terms of the time of the growth process and to determine from this the fractal dimension furthermore convergence of the boundary process to a diffusion process is shown leading to an interesting duality relation between the growth process and a corresponding branching process

a phase transition in random coin tossing suppose that a coin with bias theta is tossed at renewal times of a renewal process and a fair coin is tossed at all other times let mu theta be the distribution ofthe observed sequence of coin tosses and let u n denote the chance of a renewal at time n harris and keane showed that if sum n 1 infty u n 2 infty then mu theta and mu 0 are singular while if sum n 1 infty u n 2 infty and theta is small enough then mu theta is absolutely continuous with respect to mu 0 they conjectured that absolute continuity should not depend on theta but only on the square summability of u n we show that in fact the power law governing the decay of u n is crucial and for some renewal sequences u n there is a phase transition at a critical parameter theta c in 0 1 for theta theta c the measures mu theta and mu 0 are mutually absolutely continuous but for theta theta c they are singular we also prove that when u n o n 1 the measures mu theta for theta in 1 1 are all mutually absolutely continuous

the galton board limit theorems and recurrence we study a particle moving in r2 under a constant external force and bouncing off a periodic array of convex domains scatterers the latter must satisfy a standard finite horizon condition to prevent ballistic collision free motion this model is known to physicists as galton board it is also identical to a periodic lorentz gas previous heuristic and experimental studies have suggested that the particle s speed v t should grow as t1 3 and its coordinate x t as t2 3 we prove these conjectures rigorously we also find limit distributions for the rescaled velocity t 1 3v t and position t 2 3x t in addition quite unexpectedly we discover that the particle s motion is recurrent that means that a ball dropped on an idealized galton board will roll down but from time to time it should bounce all the way back up with probability one

analysis and design of integral sliding manifolds for systems with unmatched perturbations the robustness properties of integral sliding mode controllers are studied this note shows how to select the projection matrix in such a way that the euclidean norm of the resulting perturbation is minimal it is also shown that when the minimum is attained the resulting perturbation is not amplified this selection is particularly useful if integral sliding mode control is to be combined with other methods to further robustify against unmatched perturbations h sub spl infin is taken as a special case simulations support the general analysis and show the effectiveness of this particular combination

output feedback variable structure control design this paper describes a procedure for variable structure output feedback control vsofc for a class of multivariable linear time invariant systems the equivalent control method of utkin in the output feedback mode yields a reduced order system exhibiting output feedback equivalent dynamics using the kimura davison sufficient conditions for pole placement using output feedback a sufficient condition is derived to assign the eigenvalues of the reduced order system the observability and controllability of the reduced system is discussed two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques a control design is discussed as well

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

simulating sensitivities of conditional value at risk conditional value at risk cvar is both a coherent risk measure and a natural risk statistic it is often used to measure the risk associated with large losses in this paper we study how to estimate the sensitivities of cvar using monte carlo simulation we first prove that the cvar sensitivity can be written as a conditional expectation for general loss distributions we then propose an estimator of the cvar sensitivity and analyze its asymptotic properties the numerical results show that the estimator works well furthermore we demonstrate how to use the estimator to solve optimization problems with cvar objective and or constraints and compare it to a popular linear programming based algorithm

full order observers for linear systems with unknown inputs this note presents a simple method to design a full order observer for linear systems with unknown inputs the necessary and sufficient conditions for the existence of the observer are given

the capacity of finite state markov channels with feedback we consider a class of finite state markov channels with feedback we first introduce a simplified equivalent channel model and then construct the optimal stationary and nonstationary input processes that maximize the long term directed mutual information furthermore we give a sufficient condition under which the channel s shannon capacity can be achieved by a stationary input process the corresponding converse coding theorem and direct coding theorem are proved

a long time tail for random walk in random scenery consider a simple random walk on d whose sites are colored black or white independently with probabilityq resp 1 q walk and coloring are independent letn k be the number of steps by the walk between itskth and k 1 th visits to a black site i e the length of itskth white run and let k e n k q 1 our main result is a proof that lim k k d 2 k 1 q q d 2 2 d 2 d 2 since it is known thatq 1 k e n 1 n k 1 b e n 1 b e n k 1 b withb the event that the origin is black exhibits a long time tail in the run length autocorrelation function numerical calculations of k 1 k 100 ind 1 2 and 3 show that there is an oscillatory behavior of k for smallk this damps exponentially fast following which the power law sets in fairly rapidly we prove that if the coloring is not independent but is convex in the sense of fkg then the decay of k cannot be faster than

further results on variable output feedback controllers zak and hui 1993 proposed a sliding mode controller for linear multiple input multiple output mimo systems using static output feedback the author s previous paper 1996 provides an improvement of the output feedback controller of zak hui for a class of linear single input single output siso systems that eliminated two important limitations 1 system uncertainties must be bounded by the system output and 2 a requirement of a matrix inequality the controller developed previously can guarantee global closed loop stability this paper extends the previous results to linear mimo systems it is emphasized that the proposed mimo controller yields global closed loop stability whereas the one in zak hui can only guarantee local stability an application of the proposed mimo controller to an aircraft model is included to show the effectiveness of the method

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

an ilmi approach to robust static output feedback sliding mode control this paper proposes a systematic methodology for designing robust static output feedback sliding mode control sofsmc for a class of systems with mismatched uncertainties this methodology consists of two parts one is a new sufficient and necessary condition for the existence problem in terms of two matrix inequalities the other is a direct solving method using the iterative linear matrix inequality ilmi technique the main advantages of this method are that only original system parameters are involved without any extra coordinate changes and that appealing to solving the static output feedback stabilization sofs problem is no longer required in addition the ilmi algorithm includes an optimal part to avoid high control efforts a numerical example also demonstrates the efficacy of the proposed approach