On almost sure convergence of amarts and martingales without the Radon-Nikodym property

It is shown here that for any Banach spaceE-valued amart (X _n) of classB, almost sure convergence off(X_n) tof(X) for eachf in a total subset ofE ^* implies scalar convergence toX.     

Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

Retracts, fixed point property and existence of periodic points

Let X be a metric space, f ∈ C⁰( X), andV ⊂X. The set-trajectory ( V, f( V), …,f ⁿ (V)) is investigated and some conditions for f to have periodic points with given periods are obtained.     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator B_r,r(f) which maps the set of 2-periodic functionsf(t) X^(r) (X^(r)=C^(r) or L₁ ^(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L₁ metric, E_n(f)_X and (f, )_X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; K_r are the known Favard constants.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 21–30, July, 1973.In conclusion, the author wishes to express his deep gratitude to N. P. Korneichuk under whose guidance this paper was written.     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

On squares in polynomial products

Let f(X) ? \mathbb Z[X]{f(X) \in \mathbb {Z}[X]} be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the product

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809     

Mixture Dynamics and Regime Switching Diffusions with Application to Option Pricing

In this paper we present a class of regime switching diffusion models described by a pair $(X(t), Y(t)) \in \mathbb{R}^n \times {\cal S}In this paper we present a class of regime switching diffusion models described by a pair (X(t), Y(t)) ? \mathbbRⁿ ×S(X(t), Y(t)) \in \mathbb{R}^n \times {\cal S}, S = {1,2,?, N }{\cal S} = \{1,2,\ldots, N \}, Y(t) being a Markov chain, for which the marginal probability of the diffusive component X(t) is a given mixture. Our main motivation is to extend to a multivariate setting the class of mixture models proposed by Brigo and Mercurio in a series of papers. Furthermore, a simple algorithm is available for simulating paths through a thinning mechanism. The application to option pricing is considered by proposing a mixture version for the Margrabe Option formula and the Heston stochastic volatility formula for a plain vanilla.     

On solution sets of multivalued differential equations

Let X be a Banach space, 2^x\? the nonempty subsets of X,J = [o,a]?R and F:J×X→2^x\? a multivalued map. We consider U′ ? F(t,u) a.e. on J, u(o) = X_p ? X. A solution of (1) is understood to be a.e. differentiable with u′ Bochner integrable over J such that u(t) =X₀ + ∫^{0 t} u′(s)ds on J and u′(t)?F(t,u(t)) a.e. Under appropriate conditions on F the set S of solutions to (1) is compact ≠ ? in C_X (J), the space of continuous v : J → X with ∣v∣₀ = max∣v(t)∣. We concentrate on maps F with F(t,.) upper semicontinuous andshow that S is connected or even a compact R_δ in the sense of Borsuk. This is interesting in itself, but also in connection with the multivalued Poincare map in case F is periodic in time.     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.