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This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables) - they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization - the Kendall convolution (associated with the Williamson transform), up to the most complicated here - Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

The classical Cramer-Lundberg model was the first attempt to describe the financial condition of the insurance company. The incomes were approximated by a steady stream of money, insurance payments were not limited and could take any value from zero to infinity. The society did not invest any part of its money, do not have any employees, shareholders or enterprise maintenance costs. There exists many modifications of the Cramer-Lundberg model which cover at least some of the problems described here, but usually they require insight into the internal financial policy of the insurance company. We propose here another modification based on Markov processes defined by generalized convolutions. Thanks to the generalized convolutions we can approximate stochastically the internal financial policy of the company based on publicly available data. In this paper we focus on computing the ruin probability for an infinite time horizon for the Markov processes Cramer-Lundberg model where the transition probabilities are defined by generalized convolutions, in particular $\alpha$-convolution, maximal convolution and the Kendall convolution.

Exciton spin and related optical polarization in self-assembled InAs/In$_{0.53}$Ga$_{0.23}$Al$_{0.24}$As/InP(001) quantum dashes emitting at 1.55 {\mu}m are investigated by means of polarization- and time-resolved photoluminescence, as well as photoluminescence excitation spectroscopy, at cryogenic temperature. We investigate the influence of highly non-resonant and quasi-resonant optical spin pumping conditions on spin polarization and spin memory of the quantum dash ground state. We show that a spin pumping scheme, utilizing the longitudinal-optical-phonon-mediated coherent scattering process, can lead to the polarization degree above 50%. We discuss the role of intrinsic asymmetries in the quantum dash that influence values of the degree of polarization and its time evolution., Comment: 4 pages, 3 figures

We report on strongly temperature-dependent kinetics of negatively charged carrier complexes in asymmetric InAs/AlGaInAs/InP quantum dots (dashes) emitting at telecom wavelengths. The structures are highly elongated and of large volume, which results in atypical carrier confinement characteristics with $s$-$p$ shell energy splittings far below the optical phonon energy, which strongly affects the phonon-assisted relaxation. Probing the emission kinetics with time-resolved microphotoluminescence from a single dot, we observe a strongly non-monotonic temperature dependence of the charged exciton lifetime. Using a kinetic rate-equation model, we find that a relaxation side-path through the excited charged exciton triplet states may lead to such behavior. This, however, involves efficient singlet-triplet relaxation via the electron spin-flip. Thus, we interpret the results as an indirect observation of strongly enhanced electron spin relaxation without magnetic field, possibly resulting from atypical confinement characteristics., Comment: 5 pages, 5 figures

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.

We present experimental and theoretical investigation of exciton recombination dynamics and the related polarization of emission in highly in-plane asymmetric nanostructures. Considering general asymmetry- and size-driven effects, we illustrate them with a detailed analysis of InAs/AlGaInAs/InP elongated quantum dots. These offer a widely varied confinement characteristics tuned by size and geometry that are tailored during the growth process, which leads to emission in the application-relevant spectral range of 1.25-1.65 {\mu}m. By exploring the interplay of the very shallow hole confining potential and widely varying structural asymmetry, we show that a transition from the strong through intermediate to even weak confinement regime is possible in nanostructures of this kind. This has a significant impact on exciton recombination dynamics and the polarization of emission, which are shown to depend not only on details of the calculated excitonic states but also on excitation conditions in the photoluminescence experiments. We estimate the impact of the latter and propose a way to determine the intrinsic polarization-dependent exciton light-matter coupling based on kinetic characteristics., Comment: 11 pages, 8 figures

The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best tool for problems connected with the Kendall generalized convolution.

A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where $X'$ is an independent copy of $X$ and $\Theta$ is independent of $X$. This is equivalent (see [12]) with the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that $$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta, \quad \quad \quad \quad \quad (\ast) $$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define weak generalized convolution of measures defined by the formula $$ {\mathcal L}(Q_1) \otimes_{\mu} {\mathcal L}(Q_2) = {\mathcal L}(\Theta), $$ if the equation $(\ast)$ holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\mathcal L}(X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this convolution. The main result of this paper is the analog of the L\'evy-Khintchine representation theorem for $\otimes_{\mu}$-infinitely divisible distributions.

In the paper we consider a generalizations of the notion of Poisson process to the case when classical convolution is replaced by generalized convolution in the sense of K. Urbanik [16] following two classical definitions of Poisson process. First, for every generalized convolution $\diamond$ we define $\diamond$-generalized Poisson process type I as a Markov process with the $\diamond$-generalized Poisson distribution. Such processes have stationary independent increments in the sense of generalized convolution, but usually they do not live on $\mathbb{N}_0$. The $\diamond$-generalized Poisson process type II is defined as a renewal process based on the sequence $S_n$, which is a Markov process with the step with the lack of memory property. Such processes take values in $\mathbb{N}_0$, however they do not have to be Markov processes, do not have to have independent increments, even in generalized convolution sense. It turns out that the second construction is possible only for monotonic generalized convolutions which admit the existence of distributions with lack of memory, thus we also study these properties.

In this paper, we present a comprehensive theory of generalized and weak generalized convolutions, illustrate it by a large number of examples, and discuss the related infinitely divisible distributions. We consider L\'{e}vy and additive process with respect to generalized and weak generalized convolutions as certain Markov processes, and then study stochastic integrals with respect to such processes. We introduce the representability property of weak generalized convolutions. Under this property and the related weak summability, a stochastic integral with respect to random measures related to such convolutions is constructed., Comment: Published at http://dx.doi.org/10.3150/14-BEJ653 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

The electron spin coherence in n-doped and undoped, self-assembled CdSe/Zn(S,Se) quantum dots has been studied by time-resolved pump-probe Kerr rotation. Long-lived spin coherence persisting up to 13 ns after spin orientation has been found in the n-doped quantum dots, outlasting significantly the lifetimes of charge neutral and negatively charged excitons of 350 - 530 ps. The electron spin dephasing time as long as 5.6 ns has been measured in a magnetic field of 0.25 T. Hyperfine interaction of resident electrons with a nuclear spin fluctuations is suggested as the main limiting factor for the dephasing time. The efficiency of this mechanism in II-VI and III-V quantum dots is analyzed., Comment: 9 pages, 7 figures, 1 table

We study experimentally and theoretically polarization-dependent luminescence from an ensemble of quantum-dot-like nanostructures with a very large in-plane shape anisotropy (quantum dashes). We show that the measured degree of linear polarization of the emitted light increases with the excitation power and changes with temperature in a non-trivial way, depending on the excitation conditions. Using an approximate model based on the k.p theory, we are able to relate this degree of polarization to the amount of light hole admixture in the exciton states which, in turn, depends on the symmetry of the envelope wave function. Agreement between the measured properties and theory is reached under assumption that the ground exciton state in a quantum dash is trapped in a confinement fluctuation within the structure and thus localized in a much smaller volume of much lower asymmetry than the entire nanostructure., Comment: 13 pages, 9 figures; considerably extended, additional discussion and new figures included

In this paper we give a first attempt to define and study stable distributions with respect to the weak generalized convolution, focusing our attention on the symmetric weakly stable distribution. As in the case of the classical convolution, characterization of distributions stable in the sense of the weak generalized convolution depends on solving some functional equations in the class of characteristic functions., Comment: 31 pages

The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a diluted hole gas has been studied by picosecond pump-probe Kerr rotation with an in-plane magnetic field. For resonant optical excitation of the positively charged exciton the spin precession shows two types of oscillations. Fast oscillating electron spin beats decay with the radiative lifetime of the charged exciton of 50 ps. Long lived spin coherence of the holes with dephasing times up to 650 ps. The spin dephasing time as well as the in-plane hole g factor show strong temperature dependence, underlining the importance of hole localization at cryogenic temperatures., Comment: 5 pages, 4 figures in PostScript format

We study the energy structure of two-dimensional holes in p-type single Al_{1-x}Ga_{x}As/GaAs heterojunctions under a perpendicular magnetic field. Photoluminescence measurments with low densities of excitation power reveal rich spectra containing both free and bound-carrier transitions. The experimental results are compared with energies of valence-subband Landau levels calculated using a new numerical procedure and a good agreement is achieved. Additional lines observed in the energy range of free-carrier recombinations are attributed to excitonic transitions. We also consider the role of many-body effects in photoluminescence spectra., Comment: 13 pages, 10 figures, accepted to Physical Review B