Volume 17, Issue 4

In the present paper we construct an example of a quaternion random variable such that Polya's type characterization theorem of Gaussian distributions does not hold. The matter is that in the linear form, consisting of the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and the other part on the left-hand side. This gives a negative answer to the question posed in [Vakhania and Chelidze, Teor. Veroyatnost. i Primenen. 54: 337–344, 2009].

We give a Mazur–Ulam type theorem for Riesz spaces. In particular, a generalization of the Mazur–Ulam theorem is given in terms of a lattice normed space.

In this note, we give a limit theorem related to the notion of regular variation for functions defined on symmetric matrices. We consider the problem under the condition of invariance by the orthogonal group, and under the invariance by the subgroup of the orthogonal group corresponding to diagonal matrices.

We discuss certain relationships between the following fundamental concepts of analysis: measurability and continuity. Some examples are presented, which underline deep connections between measurable and continuous real-valued functions. In particular, a variant of Luzin's C -property is formulated for a class of measures significantly wider than the class of Borel measures. In this context, absolutely nonmeasurable functions and Sierpiński–Zygmund type functions are considered and compared.

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions on the unit disk satisfy the inequality for all n > 2, with the equality only for the Koebe function. This conjecture remained open for n > 3. We provide here the proof of this inequality for n = 4, 5, 6. It relies on the holomorphic homotopy of univalent functions and comparison of generated singular conformal metrics in the disk. The extremality of Koebe's function follows from hyperbolic properties.

The recursive estimation problem of a one-dimensional parameter in the trend coefficient of a diffusion process is considered. The asymptotic properties of recursive estimators are derived, based on the results on the asymptotic behaviour of a Robbins–Monro type SDE. Various special cases are considered.

We study utility maximization problem for general utility functions using the dynamic programming approach. An incomplete financial market model is considered, where the dynamics of asset prices is described by an -valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic partial differential equation related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward stochastic differential equation. The cases of power, exponential and logarithmic utilities are considered as examples.

We consider the problem of statistical estimation of the logarithmic derivative of a measure in an infinite-dimensional Hilbert space. It is shown that an approximating sequence of finite-dimensional estimates can be constructed for the unknown logarithmic derivative using independent observation data.

We propose new scale-invariant tests for exponentiality based on the characterization in terms of order statistics. Limiting distributions and large deviations of new statistics are described and their local Bahadur efficiency for common alternatives is calculated.

Universal (pointwise uniform and time shifted) truncation error upper bounds are presented for the Whittaker–Kotel'nikov–Shannon sampling restoration sum for Bernstein function classes , q > 1, d ∈ ℕ, when the decay rate of the sample functions is unknown. The case of regular sampling is discussed. Extremal properties of the related series of sinc functions are investigated.

For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖ A ‖ < 1, I be the identity operator and 𝔄 c ≔ { A ∈ 𝔄 : ‖ I – A ‖ ≤ c (1 – ‖ A ‖)}, where c ≥ 1 is a constant. Let, moreover, ( x k ) k ≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ { I } → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ { I } is continuous at I .