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For the moving average process $X_n=\rho \xi_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\ge -1}$ is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$, for $N\to\infty$. We exploit that the exponential decay rate $\lambda_\rho$ of that quantity, called the persistence exponent, is given by the leading eigenvalue of a concrete integral operator. This makes it possible to study the problem with purely functional analytic methods. In particular, using methods from perturbation theory, we show that the persistence exponent $\lambda_\rho$ can be expressed as a power series in $\rho$. Finally, we consider the persistence problem for the Slepian process, transform it into the moving average setup, and show that our perturbation results are applicable., Comment: 27 pages

We consider the occupation area of spherical (fractional) Brownian motion, i.e. the area where the process is positive, and show that it is uniformly distributed. For the proof, we introduce a new simple combinatorial view on occupation times of stochastic processes that turns out to be surprisingly effective. A sampling method is used to relate the moments of occupation times to persistence probabilities of random walks that again relate to combinatorial factors in the moments of beta distributions. Our approach also yields a new and completely elementary proof of L\'evy's second arcsine law for Brownian motion. Further, combined with Spitzer's formula and the use of Bell polynomials, we give a characterisation of the distribution of the occupation times for all L\'evy processes.

Popular methods to quantify transmitted data quality are the Age of Information (AoI), the Query Age of Information (QAoI), and the Age of Incorrect Information (AoII). We consider these metrics in a point-to-point wireless communication system, where the transmitter monitors a process and sends status updates to a receiver. The challenge is to decide on the best time for an update, balancing the transmission energy and the age-based metric at the receiver. Due to the inherent risk of high age-based metric values causing complications such as unstable system states, we introduce the new concept of risky states to denote states with high age-based metric. We use this new notion of risky states to quantify and minimize this risk of experiencing high age-based metrics by directly deriving the frequency of risky states as a novel risk-metric. Building on this foundation, we introduce two risk-sensitive strategies for AoI, QAoI and AoII. The first strategy uses system knowledge, i.e., channel quality and packet arrival probability, to find an optimal strategy that transmits when the age-based metric exceeds a tunable threshold. A lower threshold leads to higher risk-sensitivity. The second strategy uses an enhanced Q-learning approach and balances the age-based metric, the transmission energy and the frequency of risky states without requiring knowledge about the system. Numerical results affirm our risk-sensitive strategies' high effectiveness.

We condition a Brownian motion on having an atypically small $L_2$-norm on a long time interval. The obtained limiting process is a non-stationary Ornstein-Uhlenbeck process., Comment: 16 pages

We consider the persistence probability of a certain fractional Gaussian process $M^H$ that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of $M^H$ exists and is continuous in the Hurst parameter $H$. Further, the asymptotic behaviour of the persistence exponent for $H\downarrow0$ and $H\uparrow1$, respectively, is studied. Finally, for $H\to 1/2$, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that $M^{1/2}$ vanishes., Comment: 25 pages

We show that a Brownian motion on $\mathbb{R}_{\ge 0}$ which is allowed to spend a total of $s > 0$ time units outside a bounded interval does not leave the interval at all. This can be seen as an extreme example of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behaviour of the probability that a Brownian motion on $[0,T]$ spends limited time outside a bounded interval, as $T \to \infty$., Comment: 19 pages

We show that a properly scaled stretched long Brownian chain converges to a two-parametric stochastic process, given by the sum of an explicit deterministic continuous function and the solution of the stochastic heat equation with zero boundary conditions., Comment: 15pages

Event-triggered control has shown the potential for providing improved control performance at the same average sampling rate when compared to time-triggered control. While this observation motivates numerous event-triggered control schemes, proving it from a theoretical perspective has only been achieved for a limited number of settings. Inspired by existing performance analyses for the single-loop case, we provide a first fundamental performance comparison of time- and event-triggered control in a multi-agent consensus setting. For this purpose, we consider undirected connected network topologies without communication delays, a level-triggering rule for event-triggered control, and the long-term average of the quadratic deviation from consensus as a performance measure. The main finding of our analysis is that time-triggered control provably outperforms event-triggered control beyond a certain number of agents in our particular setting. We thereby provide an illustrative distributed problem setup in which event-triggered control results in a performance disadvantage when compared to time-triggered control in the case of large networks. Moreover, we derive the asymptotic order of the performance measure under both triggering schemes which gives more insights into the cost relationship for large numbers of agents. Thus, by presenting an analysis for a particular setup, this work points out that transferring an event-triggering scheme from the single-loop to the multi-agent setting can lead to a loss of the often presumed superiority of event-triggered control over time-triggered control. In particular, the design of performant decentralized event-triggering schemes can therefore pose additional challenges when compared to the analogue single-loop case., Comment: 19 pages, 3 figures. This article builds upon our preceding work: https://doi.org/10.1109/CDC51059.2022.9993301

It is well known that the employed triggering scheme has great impact on the control performance when control loops operate under scarce communication resources. Various practical and simulative works have demonstrated the potential of event-triggered control to reduce communication while providing a similar performance level when compared to time-triggered control. For non-cooperative networked control systems, analytical performance comparisons of time- and event-triggered control support this finding under certain assumptions. While being well-studied in the non-cooperative setting, it remains unclear if and how the performance relationship of the triggering schemes is altered in a multi-agent system setup. To close this gap, in this paper, we consider a homogeneous single-integrator multi-agent consensus problem for which we compare the performance of time- and event-triggered control schemes analytically. Under the assumption of equal average triggering rates, we use the long-term average of the quadratic deviation from consensus as a performance measure to contrast the triggering schemes. Contrary to the non-cooperative setting, we prove that event-triggered control performs worse than time-triggered control beyond a certain number of agents in this setup. In addition, we derive the asymptotic order of the performance measure as a function of the number of agents under both triggering schemes., Comment: 7 pages, 1 figure, accepted: 61st IEEE Conference on Decision and Control (CDC 2022)

We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under non-negativity assumptions of covariance functions and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the single process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in Cheridito (2001) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$ where $H$ is the Hurst parameter of the underlying fractional Brownian motion., Comment: 19 pages

We condition a Brownian motion with arbitrary starting point $y \in \mathbb{R}$ on spending at most $1$ time unit below $0$ and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero $g=g^y$ and of its occupation time $\Gamma=\Gamma^y$ below $0$ as functions of $y$. This generalizes a result of Benjamini and Berestycki from 2011, which covers the special case $y=0$. Additionally, we study the behavior of the distributions of $g^y$ and $\Gamma^y$, respectively, for $y \to \pm\infty$.

We investigate the behaviour of a finite chain of Brownian particles, interacting through a pairwise potential $U$, with one end of the chain fixed and the other end pulled away, in the limit of slow pulling speed and small Brownian noise. We study the instant when and the place where the chain "breaks", that is, the distance between two neighbouring particles becomes larger than a certain threshold. We assume $U$ to be attractive and strictly convex up to the break distance, and three times continuously differentiable. We consider the regime, where both the pulling and the noise significantly influence the distribution of the break time and break position. It turns out that in this regime there is a universality of both the break time distribution and the break position distribution, in the sense that the limiting quantities do not depend on the details of $U$, but only on its curvature at the break distance., Comment: 32 pages

We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$., Comment: Minor corrections

With $\{\xi_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $\rho(i):=\mathbb{E}[ \xi_0\xi_i]$ and $\{\sigma(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell \sigma(i) \xi_i$, $\ell\ge 1$. For summable correlations $\rho$, we show that the persistence exponent is universal. On the contrary, for non-summable $\rho$, even for polynomial weight functions $\sigma(i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the polynomial rate $p$ of $\sigma$. In this case, we show existence of the persistence exponent $\theta(H,p)$ and study its properties as a function of $(p,H)$. During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations -- e.g.\ a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero -- that might be of independent interest., Comment: Minor changes, accepted in SPA

We investigate the behaviour of a finite chain of Brownian particles, interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed, in the limit of slow speed and small Brownian noise. We study the instant when the chain "breaks", that is, the distance between two neighboring particles becomes larger than a certain limit. There are three different regimes depending on the relation between the speed of pulling and the Brownian noise. We prove weak limit theorems for the break time and the break position for each regime., Comment: 60 pages

We study a modification of the fractional analogue of the Brownian meander, which is Brownian motion conditioned to be positive on the time interval ${[0,1]}$. More precisely, we determine the weak limit of a fractional Brownian motion which is penalized -- instead of being killed -- when leaving the positive half-axis. In the Brownian case, we give a representation of the limiting process in terms of an explicit SDE and compare it to the SDE fulfilled by the Brownian meander., Comment: Minor corrections

We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtm\"uller (2000), who study the same objects under the assumption of finite exponential moments, and Gantert et al.\ (2014), who study finite weighted sums with stretched exponential tails., Comment: 11 pages; main result slightly generalized w.r.t. v1

We study the asymptotics of the ruin probability in the Cram\'er-Lundberg model with a modified notion of ruin. The modification is as follows. If the portfolio becomes negative, the asset is not immediately declared ruined but may survive due to certain mechanisms. Under a rather general assumption on the mechanism - satisfied by most such modified models from the literature - we study the relation of the asymptotics of the modified ruin probability to the classical ruin probability. This is done under the Cram\'er condition as well as for subexponential integrated claim sizes., Comment: 9 pages

For AR(1)-processes $X_n=\rho X_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\in\mathbb{N}}$ is an i.i.d. sequence of random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$ for $N\to\infty$. For a wide class of Markov processes a recent result [Aurzada, Mukherjee, Zeitouni; arXiv:1703.06447; 2017] shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter $\rho$ for normally distributed AR(1)-processes., Comment: Version 2 contains an appendix that develops the relevant concepts from perturbation theory for linear operators

We study the first exit time of a multi-dimensional fractional Brownian motion from unbounded domains. In particular, we are interested in the upper tail of the corresponding distribution when the domain is parabola-shaped., Comment: 11pages

We consider random rectangles in $\mathbb{R}^2$ that are distributed according to a Poisson random measure, i.e., independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy-tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields and give statistical properties of these limits., Comment: 36 pages, 1 figure

We study the amount of information that is contained in "random pictures", by which we mean the sample sets of a Boolean model. To quantify the notion "amount of information", two closely connected questions are investigated: on the one hand, we study the probability that a large number of balls is needed for a full reconstruction of a Boolean model sample set. On the other hand, we study the quantization error of the Boolean model w.r.t. the Hausdorff distance as a distortion measure., Comment: 39 pages

We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes., Comment: 20 pages

We prove the existence of the persistence exponent $$\log\lambda:=\lim_{n\to\infty}\frac{1}{n}\log \mathbb{P}_\mu(X_0\in S,\ldots,X_n\in S)$$ for a class of time homogeneous Markov chains $\{X_i\}_{i\geq 0}$ taking values in a Polish space, where $S$ is a Borel measurable set and $\mu$ is an initial distribution. Focusing on the case of AR($p$) and MA($q$) processes with $p,q\in \mathbb{N}$ and continuous innovation distribution, we study the existence of $\lambda$ and its continuity in the parameters of the AR and MA processes, respectively, for $S=\mathbb{R}_{\geq 0}$. For AR processes with log-concave innovation distribution, we prove the strict monotonicity of $\lambda$. Finally, we compute new explicit exponents in several concrete examples., Comment: Minor changes. To appear in AIHP

We consider polling models in the sense of Takagi (MIT Press, 1986). In our case, the feature of the server is that it may be forced to wait idly for new messages at an empty queue instead of switching to the next station. We propose four different wait-and-see strategies that govern these waiting periods. We assume Poisson arrivals for new messages and allow general service and switchover time distributions. The results are formulas for the mean average queueing delay and characterisations of the cases where the wait-and-see strategies yield a lower delay compared to the exhaustive strategy., Comment: 28 pages, 1 figure

We study persistence probabilities of Hermite processes. As a tool, we derive a general decorrelation inequality for the Rosenblatt process, which is reminiscent of Slepian's lemma for Gaussian processes or the FKG inequality and which may be of independent interest. This allows to compute the persistence exponent for the Rosenblatt process. For general Hermite processes, we derive upper and lower bounds for the persistence probabilites with the conjectured persistence exponent, but with non-matching boundaries., Comment: 12 pages

We study persistence probabilities for random walks in correlated Gaussian random environment first studied by Oshanin, Rosso and Schehr. From the persistence results, we can deduce properties of critical branching processes with offspring sizes geometrically distributed with correlated random parameters. More precisely, we obtain estimates on the tail distribution of its total population size, of its maximum population, and of its extinction time.

We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper knowledge of the spectral structure of the underlying stationary sequence., Comment: 34pages

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in random sceneries, random processes in Brownian scenery, and the Matheron-de Marsily model in Z^2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of optimal order in the general case and improved lower bounds (compared to previous literature) for many processes., Comment: This version corrects some mistakes from the printed version. In particular, the assumptions of Theorem 4 needed to be changed; Theorem 20 needed an extra assumption; the proof of Theorem 11 has been changed

In a hybrid PON/xDSL access network, multiple Customer Premise Equipment (CPE) nodes connect over individual Digital Subscriber Lines (DSLs) to a drop-point device. The drop-point device, which is typically reverse powered from the customer, is co-located with an Optical Network Unit (ONU) of the Passive Optical Network (PON). We demonstrate that the drop-point experiences very high buffer occupancies when no flow control or standard Ethernet PAUSE frame flow control is employed. In order to reduce the buffer occupancies in the drop-point, we introduce two gated flow control protocols that extend the polling-based PON medium access control to the DSL segments between the CPEs and the ONUs. We analyze the timing of the gated flow control mechanisms to specify the latest possible time instant when CPEs can start the DSL upstream transmissions so that the ONU can forward the upstream transmissions at the full PON upstream transmission bit rate. Through extensive simulations for a wide range of bursty traffic models, we find that the gated flow control mechanisms, specifically, the ONU and CPE grant sizing policies, enable effective control of the maximum drop-point buffer occupancies.

We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension with Gaussian scenery. Second, we deal with sums of stationary Gaussian sequences with correlations exhibiting long-range dependence. Apart from the persistence probability we deal with the position of the maximum and the time spent on the positive half-axis by the process.

For a L\'evy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$., Comment: 21 pages, submitted

Dynamic circuits are well suited for applications that require predictable service with a constant bit rate for a prescribed period of time, such as cloud computing and e-science applications. Past research on upstream transmission in passive optical networks (PONs) has mainly considered packet-switched traffic and has focused on optimizing packet-level performance metrics, such as reducing mean delay. This study proposes and evaluates a dynamic circuit and packet PON (DyCaPPON) that provides dynamic circuits along with packet-switched service. DyCaPPON provides $(i)$ flexible packet-switched service through dynamic bandwidth allocation in periodic polling cycles, and $(ii)$ consistent circuit service by allocating each active circuit a fixed-duration upstream transmission window during each fixed-duration polling cycle. We analyze circuit-level performance metrics, including the blocking probability of dynamic circuit requests in DyCaPPON through a stochastic knapsack-based analysis. Through this analysis we also determine the bandwidth occupied by admitted circuits. The remaining bandwidth is available for packet traffic and we conduct an approximate analysis of the resulting mean delay of packet traffic. Through extensive numerical evaluations and verifying simulations we demonstrate the circuit blocking and packet delay trade-offs in DyCaPPON.

Building on the Ethernet Passive Optical Network (EPON) and Gigabit PON (GPON) standards, Next-Generation (NG) PONs (i) provide increased data rates, split ratios, wavelengths counts, and fiber lengths, as well as (ii) allow for all-optical integration of access and metro networks. In this paper we provide a comprehensive probabilistic analysis of the capacity (maximum mean packet throughput) and packet delay of subnetworks that can be used to form NG-PONs. Our analysis can cover a wide range of NG-PONs through taking the minimum capacity of the subnetworks making up the NG-PON and weighing the packet delays of the subnetworks. Our numerical and simulation results indicate that our analysis quite accurately characterizes the throughput-delay performance of EPON/GPON tree networks, including networks upgraded with higher data rates and wavelength counts. Our analysis also characterizes the trade-offs and bottlenecks when integrating EPON/GPON tree networks across a metro area with a ring, a Passive Star Coupler (PSC), or an Arrayed Waveguide Grating (AWG) for uniform and non-uniform traffic. To the best of our knowledge, the presented analysis is the first to consider multiple PONs interconnected via a metro network., Comment: Technical Report, School of Electrical, Computer, and Energy Eng., Arizona State University, Tempe, AZ

Current Gigabit-class passive optical networks (PONs) evolve into next-generation PONs, whereby high-speed 10+ Gb/s time division multiplexing (TDM) and long-reach wavelength-broadcasting/routing wavelength division multiplexing (WDM) PONs are promising near-term candidates. On the other hand, next-generation wireless local area networks (WLANs) based on frame aggregation techniques will leverage physical layer enhancements, giving rise to Gigabit-class very high throughput (VHT) WLANs. In this paper, we develop an analytical framework for evaluating the capacity and delay performance of a wide range of routing algorithms in converged fiber-wireless (FiWi) broadband access networks based on different next-generation PONs and a Gigabit-class multi-radio multi-channel WLAN-mesh front-end. Our framework is very flexible and incorporates arbitrary frame size distributions, traffic matrices, optical/wireless propagation delays, data rates, and fiber faults. We verify the accuracy of our probabilistic analysis by means of simulation for the wireless and wireless-optical-wireless operation modes of various FiWi network architectures under peer-to-peer, upstream, uniform, and nonuniform traffic scenarios. The results indicate that our proposed optimized FiWi routing algorithm (OFRA) outperforms minimum (wireless) hop and delay routing in terms of throughput for balanced and unbalanced traffic loads, at the expense of a slightly increased mean delay at small to medium traffic loads., Comment: Technical Report, School of Electrical, Computer, and Energy Eng. Arizona State University, Tempe

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $\alpha$-stable L\'evy processes with $\alpha<1$. Our main result states that if the left tail of the L\'evy measure is regularly varying with index $- \alpha$ and the moving boundary is equal to $1 - t^{\gamma}$ for some $\gamma<1/\alpha$, then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary $1 + t^{\gamma}$ with $\gamma<1/\alpha$ under the assumption of a regularly varying right tail with index $- \alpha$., Comment: correction of earlier mistakes, in particular the result can only be claimed for \alpha<1, to appear in: Journal of Theoretical Probability

We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM $X$ stays below a certain level until time $T$. Recently, Oshanin et al. study a physical model where persistence properties of FBM are shown to be related to scaling properties of a quantity $J_N$, called steady-state current. It turns out that for this analysis it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to a FBM without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in Oshanin et al. Moreover, we complement their findings by considering the continuous-time version of $J_N$. Finally, we use the results for moving boundaries in order to improve estimates by Molchan concerning the persistence properties of other quantities of interest, such as the time when a FBM reaches its maximum on the time interval $(0,1)$ or the last zero in the interval $(0,1)$., Comment: 13 pages

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer models and proves a conjecture by Caravenna and Deuschel., Comment: 26 pages

This article deals with the asymptotic behaviour as $t\to +\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behaviour is of the type $P[T > t]=t^{-\theta + o(1)}$ for a known or unknown positive parameter $\theta$ which is called a persistence exponent. The problem is well understood for random walks or L\'evy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field., Comment: survey paper, submitted

We revisit a result of Uchiyama (1980): given that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary $f$ is asymptotically the same as for the constant boundary. The integral test for $f$ is also necessary in some sense. After Uchiyama's result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, half-page proof for the case of a decreasing boundary. Further, we identify that the integral test is related to a repulsion effect of the three-dimensional Bessel process. Our proof gives some hope to be generalized to other processes such as FBM., Comment: 5 pages

We study the asymptotic behaviour of the tail of the distribution of the first passage time of a L\'evy process over a one-sided moving boundary. Our main result states that if the boundary behaves as $t^{\gamma}$ for large $t$ for some $\gamma<1/2$ then the probability that the process stays below the boundary behaves asymptotically as in the case of a constant boundary. We do not have to assume Spitzer's condition in contrast to all previously known results. Both positive ($+t^\gamma$) and negative ($-t^\gamma$) boundaries are considered. These results extend the findings of Greenwood and Novikov (1986) and are also motivated by results in the case of Brownian motion, for which the above result was proved in Uchiyama (1980)., Comment: 31 pages

We consider moments of the return times (or first hitting times) in a discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K. L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially -- neither globally nor locally -- fulfill the above statement., Comment: 9 pages