Invariance Principle Under Self-Normalization for Nonidentically Distributed Random Variables

Let V _n ^–1 _n be the adaptive process of self-normalized partial sums S _k of independent random variables X _i , defined by linear interpolation between the points (V _k ²/V _n ²,S _k /V _n ), kn, where V _k ²= _ik X _i ². We prove that if the X _k 's are symmetric, V _n ^–1 _n converges weakly to the Brownian motion W in each Hölder space supporting W if and only if V _n ^–1 max _kn |X _k |=o _P (1). We give some partial extension to the non symmetric case.     

Detecting Mass Points of Representing Measures

We discuss a possibility of deciding whether measures representing a moment sequence or realizing orthogonality of polynomials have atoms. This is done on the real line and in several variables.     

Three point covers with bad reduction

We study Galois covers of the projective line branched at three points with bad reduction to characteristic , under the condition that strictly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at .

     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

One-armed bandit models with continuous and delayed responses

One-armed bandit processes with continuous delayed responses are formulated as controlled stochastic processes following the Bayesian approach. It is shown that under some regularity conditions, a Gittins-like index exists which is the limit of a monotonic sequence of break-even values characterizing optimal initial selections of arms for finite horizon bandit processes. Furthermore, there is an optimal stopping solution when all observations on the unknown arm are complete. Results are illustrated with a bandit model having exponentially distributed responses, in which case the controlled stochastic process becomes a Markov decision process, the Gittins-like index is the Gittins index and the Gittins index strategy is optimal. Acknowledgement.We thank an anonymous referee for constructive and insightful comments, especially those related to the notion of the Gittins index.Both authors are funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada.     

On the Extension of Isometries between Unit Spheres of E and C（Ω）

In this paper,we study the extension of isometries between the unit spheres of some Banach spaces E and the spaces C(Ω). We obtain that if the set sm.S1(E) of all smooth points of the unit sphere S1(E) is dense in S1(E),then under some condition,every surjective isometry V0 from S1(E) onto S1(C(Ω)) can be extended to be a real linearly isometric map V of E onto C(Ω).From this resultwe also obtain some corollaries. This is the first time we study this problem on different typical spaces,and the method of proof is also very different too.     

When the “Bull” Meets the “Bear”—A First Passage Time Problem for a Hidden Markov Process

Let _t be a continuous Markov chain on N states. Consider adjoining a Brownian motion with this Markov chain so that the drift and the variance take different values when _t is in different states. This new process Z_t is a hidden Markov process. We study the probability distribution of the first passage time for Z_t.Our result, when applied to the stock market, provides an explicit mathematical interpretation of the fact that in finite time, there is positive probability for the bull (bear) market to become bear (bull).     

Continuation method for -sublinear mappings

Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies

for all , , where . Thompson's metric is among the main tools we are using.     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.