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

u statistic processes a martingale approach for i i d data x 1 x n and a kernel h the associated u statistic process is defined as formule mathematique variants of these processes occur for example in the representation of the product limit estimator ofa lifetime distribution for censored truncated data or in trimmed u statistics we derive an almost sure representation of u n under weak moment assumptions on h proofs rely on a proper decomposition of the remainder term into strong two parameter martingales

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

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

bounds on the tail probability of u statistics and quadratic forms the final version of this paper appears in bulletin of american mathematical society 31 1994 223 227 print

maximum likelihood estimation for stationary point processes in this paper we derive the log likelihood function for point processes in terms of their stochastic intensities by using the martingale approach for practical purposes we work with an approximate log likelihood function that is shown to possess the usual asymptotic properties of a log likelihood function the resulting estimates are strongly consistent and asymptotically normal under some regularity conditions as a by product a strong law of large numbers and a central limit theorem for martingales in continuous times are derived

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

a dichotomy theorem for circular colouring reconfiguration abstract let p and q be positive integers with p q 2 the reconfiguration problem for circular colourings asks given two p q colourings f and g of a graph g is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a p q colouring we show that this problem can be solved in polynomial time for 2 p q 4 and that it is pspace complete for p q 4 this generalizes a known dichotomy theorem for reconfiguring classical graph colourings as an application of the reconfiguration algorithm we show that graphs with fewer than k 1 2 cycles of length divisible by k are k colourable

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

maximal inequalities for degenerate u processes with applications to optimization estimators maximal inequalities for degenerate u processes of order k k geq 1 are established the results rest on a moment inequality due to bonami for k th order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes rates of uniform convergence are obtained the maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having u process structure as an application a semiparametric regression estimator that maximizes a u process of order 3 is shown to be sqrt n consistent and asymptotically normally distributed

on local u statistic processes and the estimation of densities of functions of several sample variables a notion of local u statistic process is introduced and central limit theorems in various norms are obtained for it this involves the development of several inequalities for u processes that may be useful in other contexts this local u statistic process is based on an estimator of the density of a function of several sample variables proposed by frees j amer statist assoc 89 1994 517 525 and as a consequence uniform in bandwidth central limit theorems in the sup and in the lp norms are obtained for these estimators

parameterized complexity of the list coloring reconfiguration problem with graph parameters abstract let g be a graph such that each vertex has its list of available colors and assume that each list is a subset of the common set consisting of k colors for two given list colorings of g we study the problem of transforming one into the other by changing only one vertex color assignment at a time while at all times maintaining a list coloring this problem is known to be pspace complete even for bounded bandwidth graphs and a fixed constant k in this paper we study the fixed parameter tractability of the problem when parameterized by several graph parameters we first give a fixed parameter algorithm for the problem when parameterized by k and the modular width of an input graph we next give a fixed parameter algorithm for the shortest variant which computes the length of a shortest transformation when parameterized by k and the size of a minimum vertex cover of an input graph as corollaries of these two results we show that the problem for cographs and the shortest variant for split graphs are fixed parameter tractable even when only k is taken as a parameter on the other hand we prove that the problem is w 1 hard when parameterized only by the size of a minimum vertex cover of an input graph

contraction and decoupling inequalities for multilinear forms and u statistics we prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space we use various means of comparison including rearrangement invariant norms e g orlicz and lorentz norms tail distributions tightness hypercontractivity and so forth

an edgeworth expansion for symmetric statistics we consider asymptotically normal statistics which are symmetric functions of n i i d random variables for these statistics we prove the validity of an edgeworth expansion with remainder o n 1 under cramer s condition on the linear part of the statistic and moment assumptions for all parts of the statistic by means of a counterexample we show that it is generally not possible to obtain an edgeworth expansion with remainder o n 1 without imposing additional assumptions on the structure of the nonlinear part of the statistic

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

on the complexity of reconfiguration problems reconfiguration problems arise when we wish to find a step by step transformation between two feasible solutions of a problem such that all intermediate results are also feasible we demonstrate that a host of reconfiguration problems derived from np complete problems are pspace complete while some are also np hard to approximate in contrast several reconfiguration versions of problems in p are solvable in polynomial time

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

filtered likelihood for point processes abstract point processes are widely used in finance and economics to model the timing of defaults market transactions unemployment spells births and a range of other events we develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution we establish an approximation to the likelihood and analyze the convergence and large sample properties of the associated estimators numerical results illustrate the behavior of our estimators

estimation of a low intensity filtered poisson process in additive white gaussian noise a poisson gauss process is defined as the sum of a filtered poisson process a poisson process passed through a linear filter and a white gaussian noise this model describes the electrical signal at the output of an optical measuring unit consisted of a photodetector and its following amplifying circuits the intensity of the poisson process in this model is proportional to the received optical power with the observations of a poisson gauss process three estimation problems are considered minimum mean squared error estimation of the poisson process at every fixed but arbitrary time minimum mean squared error estimation of the poisson intensity and the maximum likelihood estimation of the intensity the solutions to these problems are presented in terms of a complicated functional of the observed poisson gauss process which is hard to compute for an arbitrary value of the poisson intensity however under a low intensity regime nonlinear filtering schemes are developed to efficiently compute this functional this special case provides a signal processing framework for single photon detection a technology dedicated to measurement of low optical powers

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