martingales on riemannian manifolds with prescribed limit abstract we are given a random variable on a riemannian manifold and we study the set of manifold valued martingales converging to this variable more precisely we are interested in uniqueness and existence theorems we use stochastic calculus tools for existence we restrict ourselves to wiener probability spaces and we apply malliavin calculus the relations with some nonlinear partial differential equations are discussed

on a problem concerning spacings let un n 1 2 be a sequence of independent uniformly distributed random variables on 0 1 and si n i 1 n be the spacings induced by the order statistics of u1 un 1 denote begin array l m l n rm max min s i j n rm 1 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x i mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x n l 1 rm 0 mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x j l end array the exact distribution of m2 n is determined and based on analysis of something called fibonacci distribution the weak and almost sure convergence of the sequence m2 n are discussed furthermore the limiting distribution of ml n is determined for any fixed l and the equation p mathop lim limits n to infty rm nl n m l n n log n 1 1 is shown for some sequence ln

differentiable and analytic families of continuous martingales in manifolds with connection we prove that the derivative of a differentiable family x t a of continuous martingales in a manifold m is a martingale in the tangent space for the complete lift of the connection in m provided that the derivative is bicontinuous in t and a we consider a filtered probability space t 0 t 1 such that all the real martingales have a continuous version and a manifold m endowed with an analytic connection and such that the complexification of m has strong convex geometry we prove that given an analytic family a l a of random variable with values in m and such that l 0 x0 m there exists an analytic family a x a of continuous martingales such that x1 a l a for this we investigate the convexity of the tangent spaces t n m and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time t such that t 1 is arbitrarily small we use this construction of families of martingales in complex analytic manifolds to prove that every 1 measurable random variable with values in a compact convex set v with convex geometry in a manifold with a c1 connection is reachable by a v valued martingale

regularity for quasilinear second order subelliptic equations in this paper we study the regularity of solutions of the quasilinear equation where x x1 xm is a system of real smooth vector fields aij b c rm 1 assume that x satisfies the hormander condition and aij x z is positive definite we prove that if u s2 see section 2 is a solution of the above equation then u c

upper and lower class sequences for minimal uniform spacings letkn be thek th smallest spacing defined byx1 xn independent random variables uniformly distributed on 0 1 wherek 1 is a fixed integer letun be a sequence of positive numbers satisfyingun n2 we show that p n 2 k n leqq u n i o 0 1 when sum limits n u n k n infty infty i begin gathered if in addition u n uparrow then p n 2 k n geqq u n i o 0 1 when hfill sum limits n u n k n exp u n infty infty hfill end gathered ii

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

a data driven procedure for density estimation with some applications this paper deals with the probability density estimation using a kernel based approach where the window size of the kernel is found by a data driven procedure it is theoretically shown that under certain assumptions the estimated densities on bounded sets can be asymptotically unbiased when the width of window is obtained from the minimal spanning tree of the observed data the theoretical development initially carried out on r2 is applicable to higher dimensional spaces the results are experimentally verified on bounded sets with different types of distributions the behaviour of the estimator in the case of the unbounded set as in that for gaussian density is also experimentally seen to be good some applications of the proposed density estimation technique is demonstrated one application is the representative point detection algorithm which can be applied for data reduction and outlier rejection another application involves detection of border points of a dot pattern as well as finding a thinned version of the dot pattern

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

connectivity in one dimensional soft random geometric graphs in this paper we study the connectivity of a one dimensional soft random geometric graph rgg the graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on the distance between them we derive bounds on the probability that the graph is fully connected by analyzing key modes of disconnection in particular analytic expressions are given for the mean and variance of the number of isolated nodes and a sharp threshold established for their occurrence bounds are also derived for uncrossed gaps and it is shown analytically that uncrossed gaps have negligible probability in the scaling at which isolated nodes appear this is in stark contrast to the hard rgg in which uncrossed gaps are the most important factor when considering network connectivity

gradient estimates for some diffusion semigroups consider the semigroup p t of an elliptic diffusion we describe a simple stochastic method providing gradient estimates on p tf if n is a manifold endowed with a connection the method can also be applied to the associated nonlinear semigroup q t acting on n valued maps with a localization technique we deduce gradient estimates for real harmonic functions or n valued harmonic maps moreover the results are extended to a class of hypoelliptic diffusions

on the erdos renyi theorem for random fields and sequences and its relationships with the theory of runs and spacings we prove in this paper a law of erdos renyi type for arrays of independent and identically distributed random variables the relationships of our theorem with similar results obtained in the theory of runs and spacings are investigated applications include the evaluation of the rate of convergence of erdos renyi maxima in limiting cases and a generalization of a theorem of erdos and revesz on runs

strong bounds for multidimensional spacings let u1 u2 u n be independent random vectors uniformly distributed on 0 1 d we define the k th maximal spacing n k associated to u1 u1 as the k th maximal possible value of the length of a side of a d dimensional square block in 0 1 d which does not intersect the sample and which cannot be enlarged without doing so our main result is that n k n 1 log n 0 log2n 1 d almost surely as n other bounds are proposed for the limiting almost sure behavior of n k as n

one dimensional geometric random graphs with nonvanishing densities part i a strong zero one law for connectivity we consider a collection of n independent points which are distributed on the unit interval 0 1 according to some probability distribution function f two nodes are said to be adjacent if their distance is less than some given threshold value when f admits a nonvanishing density f we show under a weak continuity assumption on f that the property of graph connectivity for the induced geometric random graph exhibits a strong zero one law and we identify the corresponding critical scaling this is achieved by generalizing to nonuniform distributions a limit result obtained by levy for maximal spacings under the uniform distribution

strong laws for the maximal k spacing when k c log n we consider the maximal k spacing m n mathop max limits 0 underset raise0 3em hbox smash scriptscriptstyle leqslant i underset raise0 3em hbox smash scriptscriptstyle leqslant n 1 k left u ni k u ni right where u n1 u nn are the order statistics of an i i d sample of size n from the uniform distribution on 0 1 and u n0 0 u nn 1 1 the integer k is allowed to vary with n at a rate not exceeding log n we obtain laws of the iterated logarithm for all the k s in the given range for small k the methods used in the proofs are borrowed from extreme value theory for larger k the techniques are reminiscent of those used in the proof of the erdos renyi theorem

upper bounds for k th maximal spacings if kn denotes the k th maximal spacing generated by an i i d sequence of random variables uniformly distributed on 0 1 we show that for any p 3 p left nk n underset raise0 3em hbox smash scriptscriptstyle geqslant text log n left 2 text log text 2 n text log text 3 n ldots left 1 varepsilon right text log p n right k text i text o text right 1 text or 0 according as e 0 or e 0 we also obtain strong limiting bounds for the record times and inter record times of kn

a recursive method for the construction of irreducible representations of the orthogonal group o n an algorithm for the construction of o n gel fand zetlin states in an invariant vector space is developed the states are calculated recursively using a new type of coefficient of fractional parentage these coefficients are the eigenvectors of the gel fand invariants it is shown that the calculation of the gel fand invariants matrix elements can be reduced to the evaluation of a single generator at each step this algorithm provides a new approach to the calculation of the clebsh gordon coefficients and isoscalar factors for the orthogonal group and can be applied to construct a basis function with well defined orthogonal symmetry for physical systems where separation between collective motions and intrinsic motions associated with the group o n is of interest

collectivity and geometry iii the three dimensional case in the sp 6 sp 2 o 3 chain for closed shells as was indicated in previous papers general hamiltonians for systems of n particles in three dimensional space can be formulated in the enveloping algebra of the symplectic group sp 6n this group admits among others the subgroup sp 6 o n and as has been noticed by many authors collective hamiltonians can be formulated in the enveloping algebra of sp 6 so that their eigenstates can be characterized by a definite irreducible representation irrep of this group the mathematical problem is then to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of this group as well as of appropriate subgroups in the present series of papers the subgroups chosen where sp 2 o 3 as the casimir operator of sp 2 when n is formally related to the bohr mottelson vibrational hamiltonian bmvh while o 3 gives the angular momentum of the state we give an algorithm for determining these states that closely parallels the procedure followed for the bmvh programs are being developed to convert our algorithm into a computational tool for determining collective excitations in nuclei

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

collectivity and geometry ii the two dimensional case in a previous paper of this series we showed that nuclear collective behavior can be related to the symplectic geometry of the many nucleon system when we introduce in it appropriate constraints we showed in that paper that a full discussion of collective behavior in the many body system requires in a space of d dimensions the basis for the irreducible representations of the sp 2d group in the chain sp 2d sp 2 o d as well as the matrix elements of the generators of sp 2d in this basis in the present paper we implement this program fully for d 2 both because of the mathematical interest of the chain sp 4 sp 2 o 2 and as a stepping stone to the chain sp 6 sp 2 o 3 which is central to the discussion of collective motions

collectivity and geometry i general approach in the last decade an extensive literature appeared in which the microscopic collective behavior of nuclei was associated with definite irreducible representations irreps of either the o n or sp 6 groups where n a 1 and a is the number of nucleons it became clear that the two approaches are equivalent as problems with 3n degrees of freedom are characterized by a definite irrep of the group sp 6n and for its subgroup sp 6 o n the irrep of o n determines that of sp 6 and vice versa thus one can consider that collective effects appear when one introduces the constraint that in the many body system the states are restricted to a definite irrep of o n and thus also of sp 6 and the hamiltonians are in the enveloping algebra of sp 6 rather than in that of sp 6n once sp 6 becomes the paramount group of collective motions the problem is to determine the matrix elements of the generators of sp 6 in a basis characterized by irreps of its subgroups what subgroups to choose rowe and rosensteel

a two stage hybrid procedure for estimating an inverse regression function we consider a two stage procedure tsp for estimating an inverse regression function at a given point where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two we establish that the convergence rate of the second stage estimate can attain the parametric n 1 2 rate furthermore a bootstrapped variant of tsp btsp is introduced and its consistency properties studied this variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity finally the finite sample performance of btsp is studied through simulations and the method is illustrated on a data set

hammerstein system identification with the nearest neighbor algorithm the nonlinear characteristic in a hammerstein system i e a system in which a nonlinear memoryless subsystem and a linear dynamic are connected in a cascade is recovered with the nonparametric nearest neighbor regression estimate the a priori information is nonparametric both the nonlinear characteristic and the impulse response are completely unknown and can be of any form local and global properties of the estimate are examined whatever the probability density of the input signal the estimate converges at every continuity point of the characteristic as well as in the global sense we derive the asymptotic bias and variance of the proposed estimate as a result the optimal rate of convergence is established that additionally is independent of the shape of the input density results of numerical simulations are also presented

estimating covariation epps effect microstructure noise abstract this paper is about how to estimate the integrated covariance x y t of two assets over a fixed time horizon 0 t when the observations of x and y are contaminated and when such noisy observations are at discrete but not synchronized times we show that the usual previous tick covariance estimator is biased and the size of the bias is more pronounced for less liquid assets this is an analytic characterization of the epps effect we also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms including nonsynchronous trading microstructure noise and time discretization finally a two scales covariance estimator is provided which simultaneously cancels to first order the epps effect and the effect of microstructure noise the gain is demonstrated in data

laws of the iterated logarithm for order statistics of uniform spacings let x 1 x 2 cdots be a sequence of independent uniformly distributed random variables on lbrack 0 1 rbrack and let k n be the k th largest spacing induced by the order statistics of x 1 cdots x n 1 we show that lim sup nk n log n 2 log 2n 1 k quad text almost surely and lim inf nk n log n log 3n c quad text almost surely where log 2 leq c leq 0 and log j is the j times iterated logarithm