On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

One step semi-explicit methods based on the Cayley transform for solving isospectral flows

This note deals with the numerical solution of the matrix differential system Y′ = [B(t,Y), Y], Y(0) = Y₀, t ⩾ 0, where Y₀ is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. [B(t,Y),Y] = B(t,Y)Y − YB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y₀ and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y₀U^T(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY₀U^T)U, U(0) = I, t ⩾ 0, (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.     

Improving the Data Augmentation Algorithm in the Two-Block Setup

The data augmentation (DA) approach to approximate sampling from an intractable probability density f_X is based on the construction of a joint density, f_{X, Y}, whose conditional densities, f_X|Y and f_Y|X, can be straightforwardly sampled. However, many applications of the DA algorithm do not fall in this “single-block” setup. In these applications, X is partitioned into two components, X = (U, V), in such a way that it is easy to sample from f_Y|X, f_{U|V, Y}, and f_{V|U, Y}. We refer to this alternative version of DA, which is effectively a three-variable Gibbs sampler, as “two-block” DA. We develop two methods to improve the performance of the DA algorithm in the two-block setup. These methods are motivated by the Haar PX-DA algorithm, which has been developed in previous literature to improve the performance of the single-block DA algorithm. The Haar PX-DA algorithm, which adds a computationally inexpensive extra step in each iteration of the DA algorithm while preserving the stationary density, has been shown to be optimal among similar techniques. However, as we illustrate, the Haar PX-DA algorithm does not lead to the required stationary density f_X in the two-block setup. Our methods incorporate suitable generalizations and modifications to this approach, and work in the two-block setup. A theoretical comparison of our methods to the two-block DA algorithm, a much harder task than the single-block setup due to nonreversibility and structural complexities, is provided. We successfully apply our methods to applications of the two-block DA algorithm in Bayesian robit regression and Bayesian quantile regression. Supplementary materials for this article are available online.     

Exact tail asymptotics in bivariate scale mixture models

Let (X, Y) = (RU ₁, RU ₂) be a given bivariate scale mixture random vector, with R > 0 independent of the bivariate random vector (U ₁, U ₂). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y) assuming that R has distribution function in the Gumbel max-domain of attraction, and (U ₁, U ₂) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions.     

Functional characterization of Vasil’ev invariants

A cover of a manifold X is called an r-cover if any r points of X belong to a set in the cover. Let X and Y be two smooth manifolds, let Emb(X, Y) be the family of smooth embeddings X → Y, let M be an Abelian group, and let F: Emb(X, Y) → M be a functional. One says that the degree of F does not exceed r if for each finite open r-cover {U _i } _i∈I ; of X there exist functionals F _i : Emb(U _i , Y) → M, i ∈ I, such that for each a ∈ Emb(X, Y) one has

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.     

Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X ₁, Y ₁), (X ₂, Y ₂), ... and for large t the distribution of X ₁ +...+ X _N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y ₁, Y ₂,.... Similar expansions are derived for multivariate X ₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X ₁)+ ...+ f(X _N ) when N is large and nonrandom, and X _i , i=1, 2,..., is a discrete strongly mixing Markov chain.     

A note on operator Banach spaces containing a complemented copy of c0

Let X and Y Banach spaces. Two new properties of operator Banach spaces are introduced. We call these properties "boundedly closed" and "d-boundedly closed". Among other results, we prove the following one. Let U(X, Y){\cal U}(X, Y) an operator Banach space containing a complemented copy of c₀. Then we have: 1) If U(X, Y){\cal U}(X, Y) is boundedly closed then Y contains a copy of c₀. 2) If U(X, Y){\cal U}(X, Y) is d-boundedly closed, then X^* or Y contains a copy of c₀.     

Nonautonomous Stochastic Search in Global Optimization

We present a general method how to prove convergence of a sequence of random variables generated by a nonautonomous scheme of the form X _t =T _t (X _t−1,Y _t ), where Y _t represents randomness, used as an approximation of the set of solutions of the global optimization problem with a continuous cost function. We show some of its applications.     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (X _t ,Y _t ) be a pure jump Markov process, where X _t takes values in \bf R and Y _t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter. Accepted 7 December 1999     

Shy couplings

A pair (X, Y) of Markov processes on a metric space is called a Markov coupling if X and Y have the same transition probabilities and (X, Y) is a Markov process. We say that a coupling is “shy” if inf _{t ≥ 0} dist(X _t , Y _t ) >  0 with positive probability. We investigate whether shy couplings exist for several classes of Markov processes.     

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).     

A bivariate uniform autoregressive process

We define the bivariate first order stationary autoregressive process {(X _n ,Y _n )} with uniform marginal distribution where {X _n } and {Y _n } are the two stationary sequences with uniformU(0, 1) marginal distributions. We also estimate the unknown parameters of the model.     

Filtration Consistent Nonlinear Expectations and Evaluations of Contingent Claims

We will study the following problem.Let X_t,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation.     

A Remark on Self-Normalization for Dependent Random Variables

Let (X _t , t ∈ Z) be a stationary process, and let S _n = ∑_{1⩽ i⩽n} X _i . In this paper, we consider the central limit theorem for the self-normalized sequence S _n /U _n , where U _n ² = ∑_1⩽j⩽N Y _j ² , Y _j = ∑_{(j−1)m<i⩽jm} X _i , n = mN. We show how such a self-normalization works for AR(1) and MA(q) processes.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 173–183, April–June, 2005.     

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Extension of Transient Resolvents

For a submarkovian resolvent U=(U )_>0 with bounded initial kernel U ₀ on a Radon space X, we construct a minstable cone of potentials C on a compact metrizable space YX such that U extends to a subordonated (to C) resolvent on Y.