the signature of a rough path uniqueness in the context of controlled differential equations the signature is the exponential function on paths b hambly and t lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree like we extend hambly lyons result and their notion of tree like paths to the setting of weakly geometric rough paths in a banach space at the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures

hyperbolic development and inversion of signature abstract we develop a simple procedure that allows one to explicitly reconstruct any piecewise linear path from its signature the construction is based on the development of the path onto the hyperbolic space

formative processes with applications to the decision problem in set theory i powerset and singleton operators this paper introduces formative processes composed by transitive partitions given a family f of sets a formative process ending in the venn partition e of f is shown to exist sufficient criteria are also singled out for a transitive partition to model via a function from set variables to unions of sets in the partition all set literals modeled by e on the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by c e with c a specific computable function as a by product one of the core results on decidability in computable set theory is rediscovered namely the one that regards the satisfiability of unquantified set theoretic formulae involving boolean operators the singleton former and the powerset operator the method described which is able to exhibit a set solution when the answer is affirmative can be extended to solve the satisfiability problem for broader fragments of set theory

decision procedures for elementary sublanguages of set theory ix unsolvability of the decision problem for a restricted subclass of the δ0 formulas in set theory etude des procedures de decision pour les sous langages elementaires de la theorie des ensembles insolubilite du probleme de decision pour une sous classe restreinte des 0 formules de la theorie des ensembles etablissement de bornes

uniqueness of signature for simple curves abstract we propose a topological approach to the problem of determining a curve from its iterated integrals in particular we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation 1 p 2 are in fact moments of its winding number this relation allows us to prove that the signature series of a class of simple non smooth curves uniquely determine the curves this implies that outside a chordal sle null set where 0 4 the signature series of curves uniquely determine the curves our calculations also enable us to express the fourier transform of the n point functions of sle curves in terms of the expected signature of sle curves although the techniques used in this article are deterministic the results provide a platform for studying sle curves through the signatures of their sample paths

uniqueness for the signature of a path of bounded variation and the reduced path group we introduce the notions of tree like path and tree like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length we show that the equivalence classes form a group with some similarity to a free group and that in each class there is a unique path that is tree reduced the set of these paths is the reduced path group it is a continuous analogue of the group of reduced words the signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path we identify the paths with trivial signature as the tree like paths and prove that two paths are in tree like equivalence if and only if they have the same signature in this way we extend chen s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parameterisation in the general setting it is suggestive to think of this result as a non commutative analogue of the result that integrable functions on the circle are determined up to lebesgue null sets by their fourier coefficients as a second theme we give quantitative versions of chen s theorem in the case of lattice paths and paths with continuous derivative and as a corollary derive results on the triviality of exponential products in the tensor algebra

from random lines to metric spaces consider an improper poisson line process marked by positive speeds so as to satisfy a scale invariance property actually scale equivariance the line process can be characterized by its intensity measure which belongs to a one parameter family if scale and euclidean invariance are required this paper investigates a proposal by aldous namely that the line process could be used to produce a scale invariant random spatial network sirsn by means of connecting up points using paths which follow segments from the line process at the stipulated speeds it is shown that this does indeed produce a scale invariant network under suitable conditions on the parameter in fact it then produces a parameter dependent random geodesic metric for dd dimensional space d 2d 2 where geodesics are given by minimum time paths moreover in the planar case it is shown that the resulting geodesic metric space has an almost everywhere unique geodesic property that geodesics are locally of finite mean length and that if an independent poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region it is an open question whether the result is a sirsn in aldous sense so placing finite mean length in each compact region but it may be called a pre sirsn

on the exceedance point process for a stationary sequence it is known that the exceedance points of a high level by a stationary sequence are asymptotically poisson as the level increases under appropriate long range and local dependence conditions when the local dependence conditions are relaxed clustering of exceedances may occur based on poisson positions for the clusters in this paper a detailed analysis of the exceedance point process is given and shows that under wide conditions any limiting point process for exceedances is necessarily compound poisson more generally the possible random measure limits for normalized exceedance point processes are characterized sufficient conditions are also given for the existence of a point process limit the limiting distributions of extreme order statistics are derived as corollaries

general rough integration levy rough paths and a levy kintchine type formula we consider rough paths with jumps in particular the analogue of lyons extension theorem and rough integration are established in a jump setting offering a pathwise view on stochastic integration against cadlag processes a class of levy rough paths is introduced and characterized by a sub ellipticity condition on the left invariant diffusion vector fields and a certain integrability property of the carnot caratheodory norm with respect to the levy measure on the group using hunt s framework of lie group valued levy processes examples of levy rough paths include a standard multi dimensional levy process enhanced with a stochastic area as constructed by d williams the pure area poisson process and brownian motion in a magnetic field an explicit formula for the expected signature is given

undecidability results for restricted universally quantified formulae of set theory the problem of establishing whether there are sets satisfying a formula in the first order set theoretic language based on which involves only restricted quantifiers and has an equivalent prenex form 0 formula is neither decidable nor semidecidable in fact given any model of zf fa where fa denotes the foundation axiom the set of existential closures of 0 formulae true in the model is a productive set undecidability arises even when dealing with restricted universal quantifiers only provided a predicate is a pair x meaning that x is a pair of distinct sets is added to if satisfiability refers to models of zf fa in which a form of boffa s antifoundation axiom holds then semidecidability fails as well in fact given any such model the set of existential closures of formulae involving only restricted quantifiers and the predicate is a pair which are true in it is a productive set these results are all proved by making use of appropriate codings of turing machine computations in the set theoretic language

reconstruction for the signature of a rough path recently it was proved that the group of rough paths modulo tree like equivalence is isomorphic to the corresponding signature group through the signature map s a generalized notion of taking iterated path integrals however the proof of this uniqueness result does not contain any information on how to see the trajectory of a tree reduced rough path from its signature and a constructive understanding on the uniqueness result in particular on the inverse of s has become an interesting and important question the aim of this paper is to reconstruct a rough path from its signature in an explicit and universal way

the random difference equation x sb n a sb nx sb n 1 b sb n in the critical case let b n a n n 1 be a sequence of i i d random variables with values in r d x r the markov chain on r d which satisfies the random equation x n a n x n 1 b n is studied when e log a 1 0 no density assumption on the distribution of b 1 a 1 is made the main results are recurrence of the markov chain x n stability properties of the paths existence and uniqueness of a radon invariant measure and a limit theorem for the occupation times the results rely on a renewal theorem for the process x n a n a 1

techniques of computable set theory with applications to proof verification the original goal was the design and implementation of a proof verifier based on the very expressive formalism of set theory with a powerful inferential engine containing a wide collection of decision procedures cf 38 most of the research concentrated though on theoretical aspects namely the decision problem for fragments of set theory as shown by the numerous publications culminated in the monograph 22 which gives a complete account of the state of the art of computable set theory up to 1989 in this paper we present some recent techniques and compare them with early approaches to the decision problem in giving applications of such techniques we sort of review the decision problem in set theory and related areas with respect to the classifications of the models one is looking for namely pure set models with no individuals or stratified models and with respect to the structure of the formulae considered namely whether explicit quantification is allowed or not it must be noted however that the present survey is far from being complete for instance decision procedures relative to weak axiomatizations of set theory cf 54 55 57 62 64 65 or in presence of the anti foundation axiom are not surveyed cf 56 61 also undecidability results are not reviewed here cf 19 59 63 the paper is organized as follows section 2 introduces basic concepts about set hierarchies models realizations and the decision problem

point process and partial sum convergence for weakly dependent random variables with infinite variance let j be a strictly stationary sequence of random variables with regularly varying tail probabilities we consider via point process methods weak convergence of the partial sums s n 1 n suitably normalized when j satisfies a mild mixing condition we first give a characterization of the limit point processes for the sequence of point processes n n with mass at the points j a n j 1 n where a n is the 1 n 1 quantile of the distribution of 1 then for 0 t n 0 p s n t n np 1 t n tends to a constant which can in general be different from 1 applications of our main results to self norming sums m dependent sequences and linear processes are also given

decision algorithms for elementary topology i topological syllogistics with set and map constructs connectedness and cardinality comparison in this paper we give a decision procedure for a topological syllogistic denoted 2 involving besides the standard boolean set operators set complementation and predicates membership equality also the kuratowski closure operator continuous and closed maps with the notion of direct and inverse images surjectivity and injectivity as well as point evaluation the singleton operator a predicate for comparing the cardinalities of two given sets and a predicate expressing that a given set is topologically connected several examples are also discussed 1994 john wiley sons inc

path developments and tail asymptotics of signature for pure rough paths abstract solutions to linear controlled differential equations can be expressed in terms of global iterated path integrals along the driving path this collection of iterated integrals encodes essentially all information about the underlying path while upper bounds for iterated path integrals are well known lower bounds are much less understood and it is known only relatively recently that some types of asymptotics for the n th order iterated integral can be used to recover some intrinsic quantitative properties of the path such as the length for c 1 paths in the present paper we investigate the simplest type of rough paths the rough path analogue of line segments and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the underlying path our methodology which we believe is new for this problem involves developing paths into complex semisimple lie algebras and using the associated representation theory to study spectral properties of lie polynomials under the lie algebraic development

tail asymptotics of the brownian signature the signature of a path gamma is a sequence whose n th term is the order n iterated integrals of gamma it arises from solving multidimensional linear differential equations driven by gamma we are interested in relating the path properties of gamma with its signature if gamma is c 1 then an elegant formula of hambly and lyons relates the length of gamma to the tail asymptotics of the signature we show an analogous formula for the multidimensional brownian motion with the quadratic variation playing a similar role to the length in the proof we study the hyperbolic development of brownian motion and also obtain a new subadditive estimate for the asymptotic of signature which may be of independent interest as a corollary we strengthen the existing uniqueness results for the signatures of brownian motion

on some estimates based on sample behavior near high level excursions let j be a stationary sequence of weakly dependent random variables and letm n k be thek th largest value of j 1 j n the estimation of the parameters of the asymptotic distribution ofm n k is considered using a procedure motivated by a limit theorem pertaining to the point process sum nolimits j delta j n n bar f xi j a number of statistical issues concerning the procedure including how to select the tuning parameters are addressed the second problem that we consider is the estimation of the filter of a moving average process with heavy tails in particular the investigation covers the moving average stable process motivated by ideas in rootzen 1978 our estimator uses information contained in the sample behavior of the process near the largest excursion

limit theory for the sample autocorrelations and extremes of a garch 1 1 process the asymptotic theory for the sample autocorrelations and extremes of a garch i 1 process is provided special attention is given to the case when the sum of the arch and garch parameters is close to 1 that is when one is close to an infinite variance marginal distribution this situation has been observed for various financial log return series and led to the introduction of the igarch model in such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values and the sample autocorrelations of the squared time series have nondegenerate limit distributions we discuss the consequences for a foreign exchange rate series

stratonovich s signatures of brownian motion determine brownian sample paths the signature of brownian motion in mathbb r d over a running time interval 0 t is the collection of all iterated stratonovich path integrals along the brownian motion we show that in dimension d ge 2 almost all brownian motion sample paths running up to time t are determined by their signature over 0 t

the sample autocorrelations of heavy tailed processes with applications to arch we study the sample acvf and acf of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying this includes linear and bilinear processes with regularly varying noise and arch processes their squares and absolute values we show that the distributional limits of the sample acf can be random provided that the variance of the marginal distribution is infinite and the process is nonlinear this is in contrast to infinite variance linear processes if the process has a finite second but infinite fourth moment then the sample acp is consistent with scaling rates that grow at a slower rate than the standard root n consequently asymptotic confidence bands are wider than those constructed in the classical theory we demonstrate the theory in full detail far an arch 1 process

random walks and levy processes as rough paths we consider random walks and levy processes in a homogeneous group g for all p 0 we completely characterise almost all g valued levy processes whose sample paths have finite p variation and give sufficient conditions under which a sequence of g valued random walks converges in law to a levy process in p variation topology in the case that g is the free nilpotent lie group over mathbb r d so that processes of finite p variation are identified with rough paths we demonstrate applications of our results to weak convergence of stochastic flows and provide a levy khintchine formula for the characteristic function of the signature of a levy process at the heart of our analysis is a criterion for tightness of p variation for a collection of cadlag strong markov processes

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given