l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

isoperimetry and integrability of the sum of independent banach space valued random variables we develop a new method to study the tails of a sum of independent mean zero banach space valued random variables x i i leq n it relies on a new isoperimetric inequality for subsets of a product of probability spaces in particular we prove that for p geq 1 bigg sum i leq n x i bigg p leq frac kp 1 log p bigg bigg sum i leq n x i bigg 1 max i leq n x i p bigg where k is a universal constant other optimal inequalities for exponential moments are obtained

best constants in martingale version of rosenthal s inequality the following generalization of rosenthal s inequality was proved by burkholder a 1 p s f p d ast p leq f ast p leq b p s f p d ast p for all martingales f n it is known that a p grows like sqrt p as p rightarrow infty in this paper we prove that the growth rate of b p as p rightarrow infty is p ln p

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

brief paper bayesian estimation via sequential monte carlo sampling constrained dynamic systems nonlinear and non gaussian processes with constraints are commonly encountered in dynamic estimation problems methods for solving such problems either ignore the constraints or rely on crude approximations of the model or probability distributions such approximations may reduce the accuracy of the estimates since they often fail to capture the variety of probability distributions encountered in constrained linear and nonlinear dynamic systems this article describes a practical approach that overcomes these shortcomings via a novel extension of sequential monte carlo smc sampling or particle filtering inequality constraints are imposed by accept reject steps in the algorithm the proposed approach provides samples representing the posterior distribution at each time point and is shown to satisfy the same theoretical properties as unconstrained smc illustrative examples show that results of the proposed approach are at least as accurate as moving horizon estimation but computationally more efficient and in addition the approach indicates the uncertainty associated with these estimates

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

a monte carlo approach to nonnormal and nonlinear state space modeling abstract a solution to multivariate state space modeling forecasting and smoothing is discussed we allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation the observational equation or both an adaptive monte carlo integration technique known as the gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework the methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space missing data problems including the k step ahead prediction problem also are easily incorporated into this framework we illustrate the broad applicability of our approach with two examples a problem involving nonnormal error distributions in a linear model setting and a one step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters

error bounds for a deterministic version of the glimm scheme consider the hyperbolic system of conservation laws u t f u x 0 let u be the unique viscosity solution with initial condition u 0 x bar u x and let ue be an approximate solution constructed by the glimm scheme corresponding to the mesh sizes delta x delta t o delta x with a suitable choice of the sampling sequence we prove the estimate big u ve t cdot u t cdot big bf l 1 strut l 1 o 1 cdot sqrt delta x big ln delta x big

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

solving optimization problems with diseconomies of scale via decoupling we present a new framework for solving optimization problems with a diseconomy of scale in such problems our goal is to minimize the cost of resources used to perform a certain task the cost of resources grows superlinearly as xq q 1 with the amount x of resources used we define a novel linear programming relaxation for such problems and then show that the integrality gap of the relaxation is aq where aq is the q th moment of the poisson random variable with parameter 1 using our framework we obtain approximation algorithms for the minimum energy efficient routing minimum degree balanced spanning tree load balancing on unrelated parallel machines and unrelated parallel machine scheduling with nonlinear functions of completion times problems our analysis relies on the decoupling inequality for nonnegative random variables the inequality states that ie1nxi q cq ie1nyi q where xi are independent nonnegative random variables yi are possibly dependent nonnegative random variables and each yi has the same distribution as xi the inequality was proved by de la pena in 1990 de la pena ibragimov and sharakhmetov showed that cq 2 for qi 1 2 and cq aq1 q for q 2 we show that the optimal constant is cqe aq1 q for any q 1 we then prove a more general inequality for every convex function v elv ie1nxi r elv p ie1n yi r and for every concave function el ie1n xi r el p ie1n yi r where p is a poisson random variable with parameter 1 independent of the random variables yi

a contractive metric for systems of conservation laws with coinciding shock and rarefaction curves abstract we introduce two algorithms for the construction of weak entropy admissible solutions to a class of systems of conservation laws with coinciding shock and rarefaction curves global existence uniqueness and continuous dependence are proved for all solutions obtained by our constructive procedure the generated semigroup is contractive with respect to a riemann type metric defined in terms of glimm s wave interaction functional equivalent to the usual l 1 distance

on extremal distributions and sharp l sub p bounds for sums of multilinear forms in this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables we present results providing sharp analogues of the burkholder rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r v s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms among others we obtain the following sharp inequalities e nk 1xk t 2max nk 1extk nk 1ak t for all nonnegative r v s x1 xn with e xk x1 xk 1 ak extk k 1 n 1 x003c t x003c 2undefined e nk 1xk t e t 1 max nk 1bk nk 1ask t s for all nonnegative r v s x1 xn with e xsk x1 xk 1 ask e xtk x1 xk 1 bk k 1 n 1 x003c t x003c 2undefined 0 x003c s le t 1undefined or t 2 0 x003c s le 1undefined where 1 is a poisson random variable with parameter 1 as applications new decoupling inequalities for sums of multilinear forms are presented and sharp khintchine marcinkiewicz zygmund inequalities for generalized moving averages are obtained the results can also be used in the study of a wide class of nonlinear statistics connected to problems of long range dependence and in an econometric setup in particular in stabilization policy problems and in the study of properties of moving average and autocorrelation processes the results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved

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

saturated particle filter almost sure convergence and improved resampling nonlinear stochastic dynamical systems are widely used to model physical processes in many practical applications the state variables are defined on a compact set of the state space i e they are bounded or saturated to estimate the states of systems with saturated variables the saturated particle filter spf has recently been developed this filter exploits the structure of the saturated system using a specific importance sampling distribution in this paper we investigate the asymptotic properties of the filter in particular its almost sure convergence to the true posterior pdf furthermore an improved spf is developed that uses a novel resampling procedure to overcome the practical shortcomings of the original spf we prove that this new filter also converges almost surely to the true posterior pdf both versions of the spf are presented in easy to implement algorithmic forms

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

convex saturated particle filter in many systems the state variables are defined on a compact set of the state space to estimate the states of such systems the constrained particle filters have been used with some success the performance of the standard particle filters can be improved if the measurement information is used during the importance sampling of the filtering phase it has been shown that the particles obtained in such a way approximate the true state of the system more accurately the measurement is incorporated into the filtering algorithm through a user specified detection function which aims to detect the saturation as it occurs the algorithm derived from the aforementioned principle is called the saturated particle filter spf in our previous work we have derived a complete spf framework for the class of systems with one dimensional constraints in this paper we derive a novel convex spf that extends our method to multidimensional systems with convex constraints the effectiveness of the new method is demonstrated using an illustrative example

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

order of magnitude bounds for expectations of δ2 functions of nonnegative random bilinear forms and generalized u statistics let x 1 y 1 y 2 x n y n be independent nonnegative rv s and let b ij 1 i j n be an array of nonnegative constants we present a method of obtaining the order of magnitude of e 1 i j n b ij x i x j e 1 i j n b ij x i x j for any such x i y j and b ij and any nondecreasing function on 0 with 0 0 and satisfying a 2 growth condition furthermore this technique is extended to provide the order of magnitude of e 1 i j n f j x i y j where f ij x y 1 i j n is any array of nonnegative functions for arbitrary functions g ij x y 1 i j n the aforementioned approximation enables us to identify the order of magnitude of e 1 i j n g ij x i x j whenever decoupling results and khintchine type inequalities apply such as is convex l g ij x i x j l g ji x j x i and eg ij x i x 0 for all x in the range of x j

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

limit theorems for functionals of moving averages let x n i 1 a i e n 1 where the e i are i i d with mean 0 and finite second moment and the a i are either summable or regularly varying with index 1 1 2 the sequence x n has short memory in the former case and long memory in the latter for a large class of functions k a new approach is proposed to develop both central n rate and noncentral non n rate limit theorems for s n n 1 n k x n ek x n specifically we show that in the short memory case the central limit theorem holds for s n and in the long memory case s n can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non central limit theorem respectively further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems connections with the usual hermite expansion in the gaussian setting are also explored

sharp square function inequalities for conditionally symmetric martingales let t be a conditionally symmetric martingale taking values in a hilbert space 1ei and let s i be its square function if vp is the smallest positive zero of the confluent hypergeometric function and ap is the largest positive zero of the parabolic cylinder function of parameter p then the following inequalities are sharp 3 vp iis llp 2 moreover the constants vp and ap for the cases mentioned above are also best possible for the marcinkiewicz paley inequalities for haar functions