The asymptotic minimax theorem for Bernoulli two-armed bandit problem states that minimax risk has the order N as N → ∞, where N is the control horizon, and provides the estimates of the factor. For Gaussian twoarmed bandit with unit variances of one-step incomes and close expectations, we improve the asymptotic minimax theorem as follows: the minimax risk is approximately equal to 0.637N as N ...

In this paper we present a general result concerning the convergence to stochastic integrals with non-linear integrands. The key finding represents a generalization of Chan and Wei’s (1988) Theorem 2.4. and that of Ibragimov and Phillips’ (2004) Theorem 8.2. This result is necessary for analysing the asymptotic properties of mis-specification tests, when applied to a unit root process, for whic...

It is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A 1 2 ,2 (0) := { z ∈ C | 1 2 < |z| < 2 } . The foundations of this result rely on an operator version of Rouché’s theorem and the intermediate value theorem.

This paper is concerned with a minimal resolution of the prop for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 12) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model contains all information about the deformation theory of bialgebras and related cohomology. Algebras over this m...

This theorem is false if e ^ 0. Indeed, on the unit disc, the Kâhler metric (1 — zz)~dzdz (where n is any integer ^ 3) is complete and its curvature function K satisfies K < 0 and \K(z)\ S C(d(0, z))~. (0 = origin ofC.) The next theorem and its corollary provide information about the absence of holomorphic p-forms (p ^ 1) when the manifold is positively curved. For compact M, the result was kno...

1. Introduction. The unit disc furnished with the Poincaré metric provides a model (the Poincaré model) for the hyperbolic geometry. The linear fractional transformations of the unit disc onto itself constitute the group of motions. An analogous phenomenon arises in the unit ball of any complex Hubert space when it is furnished with the Carathéodory-Reiffen metric. The purpose of this note is t...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

The classical Hurwitz theorem says that if n first “harmonics” (2n+1 Fourier coefficients) of a continuous function f(x) on the unit circle are zero, then f(x) changes sign at least 2n + 1 times. We show that similar facts and its converse hold for any function that are orthogonal to a Chebyshev system. These theorems can be extended for convex curves in d-dimensional Euclidean space. Namely, i...

From a monotonicity property, we derive several results characterizing positive harmonic functions in the unit ball in R and positive measures on the unit sphere Sn−1. Let Bn = {x ∈ Rn : |x| < 1}, n ≥ 2 be the unit ball in Rn and Sn−1 = ∂Bn be the unit sphere. From a monotonicity property, we derive several results characterizing positive harmonic functions in Bn and positive measures on Sn−1. ...