Exit Times for a Discrete Markov Additive Process

Zbigniew Palmowski; Lewis Ramsden; Apostolos Papaioannou

Home/Archive/arXiv

arXiv · 2008.06697

Exit Times for a Discrete Markov Additive Process

Zbigniew Palmowski·Lewis Ramsden·Apostolos Papaioannou

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called $\tilde{W}$ and $\tilde{Z}$ scale matrices. which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete setup follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous-time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for $\tilde{W}$, as well as its connection with the so-called occupation mass formula. In addition to the standard one and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one and two-sided `reflected' processes.

Explore related subjects

math.PR

Keep this discovery

BibTeX RIS

Zbigniew Palmowski, Lewis Ramsden, Apostolos Papaioannou. 2020-08-15. Exit Times for a Discrete Markov Additive Process. https://arxiv.org/abs/2008.06697

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗