vector quantizers with direct sum codebooks the use of direct sum codebooks to minimize the memory requirements of vector quantizers is investigated assuming arbitrary fixed partitions necessary conditions for minimum distortion codebooks are derived first for scalar codebooks assuming mean squared error distortion and then for vector codebooks and a broader class of distortion measures an iterative procedure is described for designing locally optimal direct sum codebooks both optimal and computationally efficient suboptimal encoding schemes are considered it is shown that although an optimal encoding can be implemented by a sequential encoder the complexity of implementing optimal stagewise partitions generally exceeds the complexity of an exhaustive search of the direct sum codebook it is also shown that sequential nearest neighbor encoders can be extremely inefficient the m search method is explored as one method of improving the effectiveness of suboptimal sequential encoders representative results for simulated direct sum quantizers are presented

on the tail of the branching random walk local time consider a critical branching random walk on mathbb z d d ge 1 started with a single particle at the origin and let l x be the total number of particles that ever visit a vertex x we study the tail of l x under suitable conditions on the offspring distribution in particular our results hold if the offspring distribution has an exponential moment

universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

dynamic finite state vector quantization of digital images a vector quantization vq scheme with finite memory called dynamic finite state vector quantization dfsvq is presented the encoder consists of a large codebook so called super codebook where for each input vector a fixed number of its codevectors are chosen to generate a much smaller codebook sub codebook this sub codebook represents the best matching codevectors that could be found in the super codebook for encoding the current input vector the choice for the codevectors in the sub codebook is based on the information obtained from the previously encoded blocks where directional conditional block probability histogram matrices are used in the selection of the codevectors the index of the best matching codevector in the sub codebook is transmitted to the receiver an adaptive dfsvq scheme is also proposed in which when encoding an input vector first the sub codebook is searched for a matching codevector to satisfy a pre specified waveform distortion if such a codevector is not found in tile current sub codebook then the whole super codebook is checked for a better match if a better match is found then a signaling flag along with the corresponding index of the codevector is transmitted to the receiver both the dfsvq encoder and its adaptive version are implemented experimental results for several monochrome images with a super codebook size of 256 or 512 and different sub codebook sizes are presented

an analysis and design method for a class of nonlinear systems with nested saturations abstractthis paper is concerned with the analysis and design of a class of nonlinear systems subject to nested saturations the proposed controller incorporates both an extended state observer eso which is utilised to estimate the nonlinear dynamics of the plant as well as a set of observer based feedbacks we first present analysis results for systems with nonlinear esos and show that local stabilisation can be achieved in a region including the origin then in the case that the eso is in linear form leso the conditions for determining the estimate of the domain of attraction of the resulting closed loop system are formulated as a convex optimisation problem a linear matrix inequality based algorithm is then established to design the feedback gains and the leso gain an illustrative example is given to show the effectiveness of the proposed approach

branching brownian motion with absorption and the all time minimum of branching brownian motion with drift abstract we study a dyadic branching brownian motion on the real line with absorption at 0 drift r and started from a single particle at position x 0 with k the possibly infinite total number of individuals absorbed at 0 over all time we consider the functions s x e x s k for s 0 in the regime where is large enough so that k almost surely and that the process has a positive probability of survival we show that s if and only if s 0 s 0 for some s 0 1 and we study the properties of these functions furthermore x 0 x p x k 0 is the cumulative distribution function of the all time minimum of the branching brownian motion with drift started at 0 without absorption we give descriptions of the family s s 0 s 0 through the single pair of functions 0 x and s 0 x as extremal solutions of the kolmogorov petrovskii piskunov kpp travelling wave equation on the half line through a martingale representation and as a single explicit series expansion we also obtain a precise result concerning the tail behaviour of k in addition in the regime where k 0 almost surely we show that u x t p x k t 0 suitably centred converges to the kpp critical travelling wave on the whole real line

speed and fluctuations of n particle branching brownian motion with spatial selection we consider branching brownian motion on the real line with the following selection mechanism every time the number of particles exceeds a large given number n only the n right most particles are kept and the others killed after rescaling time by log 3n we show that the properly recentred position of the lceil alpha n rceil th particle from the right alpha in 0 1 converges in law to an explicitly given spectrally positive levy process this behaviour has been predicted to hold for a large class of models falling into the universality class of the fkpp equation with weak multiplicative noise brunet et al in phys rev e 73 5 056126 2006 and is proven here for the first time for such a model

analysis and design of discrete time linear systems with nested actuator saturations abstract this paper is concerned with the analysis and design of discrete time linear systems subject to nested saturation functions by utilizing a new compact convex hull representation of the saturation nonlinearity a linear matrix inequalities lmis based condition is obtained for testing the local and global stability of the considered nonlinear system the estimation of the domain of attraction and the design of feedback gains such that the estimation of the domain of attraction for the resulting closed loop system is maximized are then converted into some lmis based optimization problems compared with the existing results on the same problems the proposed solutions are less conservative as more slack variables are introduced into the conditions a couple of numerical examples are worked out to validate the effectiveness of the proposed approach

existence of a non averaging regime for the self avoiding walk on a high dimensional infinite percolation cluster let z n be the number of self avoiding paths of length n starting from the origin on the infinite cluster obtained after performing bernoulli percolation on mathbb z d with parameter p p c mathbb z d the object of this paper is to study the connective constant of the dilute lattice limsup n rightarrow infty z n 1 n which is a non random quantity we want to investigate if the inequality limsup n rightarrow infty z n 1 n le lim n rightarrow infty mathbb e z n 1 n obtained with the borel cantelli lemma is strict or not in other words we want to know if the quenched and annealed versions of the connective constant are equal on a heuristic level this indicates whether or not localization of the trajectories occurs we prove that when d is sufficiently large there exists p 2 c p c such that the inequality is strict for p in p c p 2 c

critical random graphs diameter and mixing time let mathcal c 1 denote the largest connected component of the critical erd s renyi random graph g n frac 1 n we show that typically the diameter of mathcal c 1 is of order n1 3 and the mixing time of the lazy simple random walk on mathcal c 1 is of order n the latter answers a question of benjamini kozma and wormald these results extend to clusters of size n2 3 of p bond percolation on any d regular n vertex graph where such clusters exist provided that p d 1 1 o n 1 3

infinite canonical super brownian motion and scaling limits we construct a measure valued markov process which we call infinite canonical super brownian motion and which corresponds to the canonical measure of super brownian motion conditioned on non extinction infinite canonical super brownian motion is a natural candidate for the scaling limit of various random branching objects on mathbb z d when these objects are critical mean field and infinite we prove that icsbm is the scaling limit of the spread out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension we conjecture that it also arises as the scaling limit in various other models above the upper critical dimension such as the incipient infinite lattice tree above 8 dimensions the incipient infinite cluster for unoriented percolation above 6 dimensions uniform spanning trees above 4 dimensions and invasion percolation above 6 dimensions

local extinction versus local exponential growth for spatial branching processes let x be either the branching diffusion corresponding to the operator lu u2 u on d rd where x 0 and 0 is bounded from above or the superprocess corresponding to the operator lu u u2 on d rd with 0 and is bounded from above but no restriction on its sign let c denote the generalized principal eigenvalue for the operator l on d we prove the following dichotomy either c 0 and x exhibits local extinction or c 0 and there is exponential growth of mass on compacts of d with rate c for superdiffusions this completes the local extinction criterion obtained by pinsky ann probab 24 1996 237 267 and a recent result on the local growth of mass under a spectral assumption given by englander and turaev ann probab 30 2002 683 722 the proofs in the above two papers are based on pde techniques however the proofs we offer are probabilistically conceptual for the most part they are based on spine decompositions or immortal particle representations along with martingale convergence and the law of large numbers further they are generic in the sense that they work for both types of processes

escape regimes of biased random walks on galton watson trees we study biased random walk on subcritical and supercritical galton watson trees conditioned to survive in the transient sub ballistic regime by considering offspring laws with infinite variance we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which in this case always yield sub ballisticity this is contrary to the walk on the supercritical tree which always has some ballistic phase

control of uncertain nonlinear systems based on observers and estimators in this paper we consider a class of nonlinear dynamical systems with zero dynamics which is subject to both unknown nonparametric dynamics and external disturbances and is required to track a given reference signal by using the output feedback our controller is designed based on both the extended state observer eso and the projected gradient estimator while the eso is used to estimate the total uncertainties the projected gradient algorithm is used to estimate the nonparametric uncertainties treated as time varying parameters this method overcomes the difficulties that the traditional active disturbance rejection control adrc technique needs to have a good prior estimate for the uncertainties in the input channel the closed loop system is shown to be semi globally stable and at the same time the tracking error can be made arbitrarily small

a fuzzy vector quantization approach to image compression the use of fuzzy clustering analysis in the early stages of a vector quantization process is able to make this process less sensitive to initialization this is justified by the fact that fuzzy clustering provides a framework for the quantitative formulation of the uncertainty typically involved in a training vector space this paper proposes a fuzzy clustering based vector quantization algorithm which employs an effective vector assignment strategy for the transition from fuzzy mode where each training vector is assigned to more than one clusters to crisp mode where each training vector is assigned to only one cluster this transition is controlled by analytical conditions that are obtained by minimizing a modified objective function for the fuzzy c means algorithm the application to image compression shows that the proposed approach is able to achieve a very efficient performance while maintaining the computational capabilities of other methods reported in the literature

branching brownian motion in a strip survival near criticality we consider a branching brownian motion with linear drift in which particles are killed on exiting the interval 0 k and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible we combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near critical survival probability this allows us to deduce the existence of a quasi stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality this paper is motivated by recent work on survival of near critical branching brownian motion with absorption at the origin by aidekon and harris near critical survival probability of branching brownian motion with an absorbing barrier 2010 unpublished manuscript as well as the work of berestycki et al ann probab 41 2013 527 618 j stat phys 143 2011 833 854

an antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation the article considers linear systems subject to actuator saturation an antiwindup technique is used to enlarge the domain of attraction of the closed loop system under an a priori designed linear dynamic feedback law the design of the antiwindup compensation gain is formulated and solved as an iterative optimization problem with lmi constraints numerical examples are used to demonstrate the effectiveness of the proposed design technique

brief paper state feedback design for input saturating quadratic systems this paper proposes a method to design stabilizing state feedback control laws for nonlinear quadratic systems subject to input saturation based on a quadratic lyapunov function a modified sector condition and a particular representation for the quadratic terms synthesis conditions in a quasi lmi form are stated in a regional local context an lmi based optimization problem is then derived for computing the state feedback gains maximizing the estimate of the stability region of the closed loop system

phase transition for the ising model on the critical lorentzian triangulation the ferromagnetic ising model without external field on an infinite lorentzian triangulation sampled from the uniform distribution is considered we prove uniqueness of the gibbs measure in the high temperature region and coexistence of at least two gibbs measures at low temperature the proofs are based on the disagreement percolation method and on a variant of the peierls contour method the critical temperature is shown to be constant a s

universality for first passage percolation on sparse random graphs we consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density assuming that the degree distribution satises a uniform x 2 logx condition we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices as well the number of edges on this path or hopcount the hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous time branching process centered by a multiple of logn where the constant is the inverse of the malthusian rate of growth of the associated branching process the minimal weight converges in distribution the limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices and a gumbel random variable and thus shows a remarkably universal behavior the proofs rely on a rened coupling between the shortest path problems on these graphs and continuous time branching processes and on a poisson point process limit for the potential closing edges of shortest weight paths between the source and destination the results extend to a host of related random graph models ranging from random rregular graphs inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence

the occurrence of large values in stationary sequences a direct proof is given of the tanny 1974 result that for certain non decreasing sequences a n it is true that lim supa n 1xn 0 or with probability one for all ergodic stationary sequences x n the condition on a n is shown to be necessary for all non decreasing a n and stationary x n lim sup an 1xn lim sup an 1x n a s similar continuous time theorems are also given

predictive residual vector quantization image coding this paper presents a new vector quantization technique called predictive residual vector quantization prvq it combines the concepts of predictive vector quantization pvq and residual vector quantization rvq to implement a high performance vq scheme with low search complexity the proposed prvq consists of a vector predictor designed by a multilayer perceptron and an rvq that is designed by a multilayer competitive neural network a major task in our proposed prvq design is the joint optimization of the vector predictor and the rvq codebooks in order to achieve this a new design based on the neural network learning algorithm is introduced this technique is basically a nonlinear constrained optimization where each constituent component of the prvq scheme is optimized by minimizing an appropriate stage error function with a constraint on the overall error this technique makes use of a lagrangian formulation and iteratively solves a lagrangian error function to obtain a locally optimal solution this approach is then compared to a jointly designed and a closed loop design approach in the jointly designed approach the predictor and quantizers are jointly optimized by minimizing only the overall error in the closed loop design however a predictor is first implemented then the stage quantizers are optimized for this predictor in a stage by stage fashion simulation results show that the proposed prvq scheme outperforms the equivalent rvq operating at the same bit rate and the unconstrained vq by 2 and 1 7 db respectively furthermore the proposed prvq outperforms the pvq in the rate distortion sense with significantly lower codebook search complexity

new rates for exponential approximation and the theorems of renyi and yaglom we introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings these are applied to obtain new rates of convergence with respect to the wasserstein and kolmogorov metrics for the theorem of renyi on random sums and generalizations of it hitting times for markov chains and to obtain a new rate for the classical theorem of yaglom on the exponential asymptotic behavior of a critical galton watson process conditioned on nonextinction the primary tools are an adaptation of stein s method stein couplings as well as the equilibrium distributional transformation from renewal theory

martingale convergence and the stopped branching random walk we discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines applying a method of lyons 1997 and lyons pemantle and peres 1995 concerning size biased branching trees we establish a relationship between stopping lines and certain stopping times consequently we develop conditions under which these supermartingales are also martingales further we prove a generalization of biggins martingale convergence theorem biggins 1977a within this context

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

vector predictive coding of speech at 16 kbits s vector quantization in its simplest form may be regarded as a generalization of pcm independent quantization of each sample of a waveform to what might be called vector pcm where a block of consecutive samples a vector is simultaneously quantized as one unit in theory a performance arbitrarily close to the ultimate rate distortion limit is achievable with waveform vector quantization if the dimension of the vector k is large enough the main obstacle in effectively using vector quantization is complexity a vector quantizer of dimension k operating at a rate of r bits sample requires a number of computations on the order of k2 kr and a memory of the same order however a low dimensional vector quantizer dimensions 4 8 achieves a remarkable improvement over scalar quantization pcm consequently using the vector quantizer as a building block and imbedding it with other waveform data compression techniques may lead to the development of a new and powerful class of waveform coding systems this paper proposes and analyzes a waveform coding system adaptive vector predictive coding avpc in which a low dimensionality vector quantizer is used in an adaptive predictive coding scheme in the encoding process a locally generated prediction of the current input vector is subtracted from the current vector and the resulting error vector is coded by a vector quantizer each frame consisting of many vectors is classified into one of m statistical types this classification determines which one of m fixed predictors and of m vector quantizers will be used for encoding the current frame

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

brief paper an improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation this paper is concerned with the treatment of saturation nonlinearity existing commonly in practical control systems an improved treatment of such nonlinearity is proposed which includes the well known linear differential inclusion method for handling saturation nonlinearity as a special case to show the effectiveness of this new treatment the problem of analysis and design of linear systems subject to nested saturation function is revisited the proposed results not only complement the existing work on this problem but also are less conservative than the existing methods four numerical examples are worked out to illustrate the applicability of the established approaches

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

total progeny in killed branching random walk we consider a branching random walk for which the maximum position of a particle in the n th generation rn has zero speed on the linear scale rn n 0 as n we further remove kill any particle whose displacement is negative together with its entire descendence the size z of the set of un killed particles is almost surely finite gantert and muller in markov process relat fields 12 805 814 2006 hu and shi in ann probab 37 2 742 789 2009 in this paper we confirm a conjecture of aldous algorithmica 22 388 412 1998 and power laws and killed branching random walks that e z while mathbf e left z rm log z right infty the proofs rely on precise large deviations estimates and ballot theorem style results for the sample paths of random walks

electrical resistance of the low dimensional critical branching random walk we show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d 0 this answers a question of barlow et al commun math phys 278 385 431 2008

enhanced multistage vector quantization by joint codebook design multistage vector quantization msvq can achieve very low encoding and storage complexity in comparison to unstructured vector quantization however the conventional stage by stage design of the codebooks in msvq is suboptimal with respect to the overall performance measure the authors introduce an algorithm for the joint design of the stage codebooks to optimize the overall performance the performance improvement although modest is achieved with no effect on encoding or storage complexity and only a slight increase in design effort

the precise tail behavior of the total progeny of a killed branching random walk consider a branching random walk on the real line with a killing barrier at zero starting from a nonnegative point particles reproduce and move independently but are killed when they touch the negative half line the population of the killed branching random walk dies out almost surely in both critical and subcritical cases where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed and by critical case when this speed is zero we investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases which solves an open problem of aldous power laws and killed branching random walks http www stat berkeley edu aldous research op brw html

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

predictive vector quantization of images the purpose of this paper is to present new image coding schemes based on a predictive vector quantization pvq approach the predictive part of the encoder is used to partially remove redundancy and the vq part further removes the residual redundancy and selects good quantization levels for the global waveform two implementations of this coding approach have been devised namely sliding block pvq and block tree pvq simulations on real images show significant improvement over the conventional dpcm and tree codes using these new techniques the strong robustness property of these coding schemes is also experimentally demonstrated

lacunarity of self similar and stochastically self similar sets let k be a self similar set in rdd of hausdorff dimension d and denote by ik c l the d dimensional lebesgue measure of its 6 neighborhood we study the limiting behavior of the quantity e d d ik e i as 6 e 0 it turns out that this quantity does not have a limit in many interesting cases including the usual ternary cantor set and the sierpinski carpet we also study the above asymptotics for stochastically self similar sets the latter results then apply to zero sets of stable bridges which are stochastically self similar in the sense of the present paper and then more generally to level sets of stable processes specifically it follows that if kt is the zero set of a realvalued stable process of index a c 1 2 run up to time t then e siit e i converges to a constant multiple of the local time at 0 simultaneously for all t 0 on a set of probability one the asymptotics for deterministic sets are obtained via the renewal theorem the renewal theorem also yields asymptotics for the mean 1e ik c in the random case while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks

a central limit theorem for biased random walks on galton watson trees let mathcal t be a rooted galton watson tree with offspring distribution p k that has p 0 0 mean m kp k 1 and exponential tails consider the biased random walk x n n 0 on mathcal t this is the nearest neighbor random walk which when at a vertex v with d v offspring moves closer to the root with probability d v and moves to each of the offspring with probability 1 d v it is known that this walk has an a s constant speed tt v lim n x n n where x n is the distance of x n from the root with tt v 0 for 0 m and tt v 0 for m for all m we prove a quenched clt for x n n tt v for m the walk is positive recurrent and there is no clt the most interesting case by far is m where the clt has the following form for almost every mathcal t the ratio x nt sqrt n converges in law as n to a deterministic multiple of the absolute value of a brownian motion our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle previously such a measure was explicitly known only for 1 and the construction of appropriate harmonic coordinates

almost sure convergence for stochastically biased random walks on trees we are interested in the biased random walk on a supercritical galton watson tree in the sense of lyons ann probab 18 931 958 1990 and lyons pemantle and peres probab theory relat fields 106 249 264 1996 and study a phenomenon of slow movement in order to observe such a slow movement the bias needs to be random the resulting random walk is then a tree valued random walk in random environment we investigate the recurrent case and prove under suitable general integrability assumptions that upon the system s non extinction the maximal displacement of the walk in the first n steps divided by log n 3 converges almost surely to a known positive constant

a mermin wagner theorem for gibbs states on lorentzian triangulations we establish a mermin wagner type theorem for gibbs states on infinite random lorentzian triangulations lt arising in models of quantum gravity such a triangulation is naturally related to the distribution p of a critical galton watson tree conditional upon non extinction at the vertices of the triangles we place classical spins taking values in a torus m of dimension d with a given group action of a torus g of dimension d d in the main body of the paper we assume that the spins interact via a two body nearest neighbor potential u x y invariant under the action of g we analyze quenched gibbs measures generated by u and prove that for p almost all lorentzian triangulations every such gibbs measure is g invariant which means the absence of spontaneous continuous symmetry breaking

stochastic barbalat s lemma and its applications in the deterministic case a significant improvement on stability analysis of nonlinear systems is caused by introducing barbalat s lemma into control area after lyapunov s second method and lasalle s theorem were established this note considers the extension of barbalat s lemma to the stochastic case to this end the uniform continuity and the absolute integrability are firstly described in stochastic forms it is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily ito diffusions when it is applied to ito diffusion processes many classical results on stochastic stability are covered as special cases

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

a cascade decomposition theory with applications to markov and exchangeable cascades a multiplicative random cascade refers to a positive t martingale in the sense of kahane on the ultrametric space t 0 1 b 1 a new approach to the study of multiplicative cascades is introduced the methods apply broadly to the problems of i non degeneracy criterion ii dimension spectra of carrying sets and iii divergence of moments criterion specific applications are given to cascades generated by markov and exchangeable processes as well as to homogeneous independent cascades 1 positive t martingales positive t martingales were introduced by jean pierre kahane as the general framework for independent multiplicative cascades and random coverings although originating in statistical theories of turbulence the general framework also includes certain spin glass and random polymer models as well as various other spatial distributions of interest in both probability theory and the physical sciences for basic definitions let t be a compact metric space with borel sigmafield b and let f p be a probability space together with an increasing sequence fn n 1 2 of sub sigmafields of f a positive t martingale is a sequence qn of b f measurable non negative functions on t such that i for each t t qn t n 0 1 is a martingale adapted to fn n 0 1 ii for p a s qn n 0 1 is a sequence of borel measurable non negative real valued functions on t let m t denote the space of positive borel measures on t and suppose that qn t is a positive t martingale for m t such that q t eqn t l let n qn denote the random measure defined by qn and dqn d t qn t t t then essentially by the martingale convergence theorem one obtains a random borel measure q such that for f c t 1 1 lim n t f t qn t dt t f t q dt a s received by the editors august 18 1994 1991 mathematics subject classification primary 60g57 60g30 60g42 secondary 60k35 60d05 60j10 60g09

occupation statistics of critical branching random walks in two or higher dimensions consider a critical nearest neighbor branching random walk on the d dimensional integer lattice initiated by a single particle at the origin let g n be the event that the branching random walk survives to generation n we obtain the following limit theorems conditional on the event g n for a variety of occupation statistics 1 let v n be the maximal number of particles at a single site at time n if the offspring distribution has finite th moment for some integer a 2 then in dimensions 3 and higher v n o p n 1 if the offspring distribution has an exponentially decaying tail then v n o p log n in dimensions 3 and higher and v n o p log n 2 in dimension 2 furthermore if the offspring distribution is nondegenerate then p vn log n g n 1 for some 0 2 let m n j be the number of multiplicity j sites in the nth generation that is sites occupied by exactly j particles in dimensions 3 and higher the random variables m n j n converge jointly to multiples of an exponential random variable 3 in dimension 2 the number of particles at a typical site i e at the location of a randomly chosen particle of the nth generation is of order o p log n and the number of occupied sites is o p n log n we also show that in dimension 2 there is particle clustering around a typical site

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

martingales in self similar growth fragmentations and their connections with random planar maps the purpose of the present work is twofold first we develop the theory of general self similar growth fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework in particular we establish many to one formulas for growth fragmentations and define the notion of intrinsic area of a growth fragmentation second we identify a distinguished family of growth fragmentations closely related to stable levy processes which are then shown to arise as the scaling limit of the perimeter process in markovian explorations of certain random planar maps with large degrees which are roughly speaking the dual maps of the stable maps of le gall and miermont in ann probab 39 1 69 2011 as a consequence of this result we are able to identify the law of the intrinsic area of these distinguished growth fragmentations this generalizes a geometric connection between large boltzmann triangulations and a certain growth fragmentation process which was established in bertoin et al ann probab accepted

local limit of labeled trees and expected volume growth in a random quadrangulation exploiting a bijective correspondence between planar quadrangulations and well labeled trees we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume the limit random surface can be described in terms of a birth and death process and a sequence of multitype galton watson trees as a consequence we find that the expected volume of the ball of radius r around a marked point in the limit random surface is r4

adaptive entropy coded predictive vector quantization of images the authors consider 2 d predictive vector quantization pvq of images subject to an entropy constraint and demonstrate the substantial performance improvements over existing unconstrained approaches they describe a simple adaptive buffer instrumented implementation of this 2 d entropy coded pvq scheme which can accommodate the associated variable length entropy coding while completely eliminating buffer overflow underflow problems at the expense of only a slight degradation in performance this scheme called 2 d pvq aecq adaptive entropy coded quantization is shown to result in excellent rate distortion performance and impressive quality reconstructions of real world images indeed the real world coding results shown demonstrate little distortion at rates as low as 0 5 b pixel