On involution kernels and large deviations principles on $ \beta $-shifts

<p style='text-indent:20px;'>Consider <inline-formula><tex-math id="M2">\begin{document}$ \beta &gt; 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lfloor \beta \rfloor $\end{document}</tex-math></inline-formula> its integer part. It is widely known that any real number <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr] $\end{document}</tex-math></inline-formula> can be represented in base <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> using a development in series of the form <inline-formula><tex-math id="M6">\begin{document}$ \alpha = \sum_{n = 1}^\infty x_n\beta^{-n} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ x = (x_n)_{n \geq 1} $\end{document}</tex-math></inline-formula> is a sequence taking values into the alphabet <inline-formula><tex-math id="M8">\begin{document}$ \{0,\; ...\; ,\; \lfloor \beta \rfloor\} $\end{document}</tex-math></inline-formula>. The so called <inline-formula><tex-math id="M9">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift, denoted by <inline-formula><tex-math id="M10">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula>, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy <inline-formula><tex-math id="M11">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M12">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. Fixing a Hölder continuous potential <inline-formula><tex-math id="M13">\begin{document}$ A $\end{document}</tex-math></inline-formula>, we show an explicit expression for the main eigenfunction of the Ruelle operator <inline-formula><tex-math id="M14">\begin{document}$ \psi_A $\end{document}</tex-math></inline-formula>, in order to obtain a natural extension to the bilateral <inline-formula><tex-math id="M15">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-shift of its corresponding Gibbs state <inline-formula><tex-math id="M16">\begin{document}$ \mu_A $\end{document}</tex-math></inline-formula>. Our main goal here is to prove a first level large deviations principle for the family <inline-formula><tex-math id="M17">\begin{document}$ (\mu_{tA})_{t&gt;1} $\end{document}</tex-math></inline-formula> with a rate function <inline-formula><tex-math id="M18">\begin{document}$ I $\end{document}</tex-math></inline-formula> attaining its maximum value on the union of the supports of all the maximizing measures of <inline-formula><tex-math id="M19">\begin{document}$ A $\end{document}</tex-math></inline-formula>. The above is proved through a technique using the representation of <inline-formula><tex-math id="M20">\begin{document}$ \Sigma_\beta $\end{document}</tex-math></inline-formula> and its bilateral extension <inline-formula><tex-math id="M21">\begin{document}$ \widehat{\Sigma_\beta} $\end{document}</tex-math></inline-formula> in terms of the quasi-greedy <inline-formula><tex-math id="M22">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-expansion of <inline-formula><tex-math id="M23">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> and the so called involution kernel associated to the potential <inline-formula><tex-math id="M24">\begin{document}$ A $\end{document}</tex-math></inline-formula>.</p>

Large Deviations for the Single-Server Queue and the Reneging Paradox

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Run-and-Tumble Motion: The Role of Reversibility

We study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distributions whose tail is in the intermediate heavy-tailed regime, which includes Weibull and lognormal type tails. The study derives asymptotic approximations for tail events of random walks. Consequently, a full large deviations principle is obtained under, essentially, minimal assumptions. As an application, an optimisation method for a large class of Quota Share (QS) risk sharing schemes used in insurance and finance is obtained.