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Frank Aurzada

Publications and source records attributed to Frank Aurzada.

At least 19 recordsLinked to original sources
arXiv2026

MA(1) processes with Laplace innovations conditioned to stay positive

We study a moving-average process with (not necessarily symmetric) Laplace innovations under the constraint of positivity. In the three nondegenerate parameter regimes $0<θ<1$, $θ>1$, and $θ<0$, we prove convergence of the conditioned finite-dimensional distributions and identify the limit as a Doob $h$-transform. The regimes lead to qualitatively different limiting dynamics: the invariant law is supported on the positive half-line for $0<θ<1$, the conditioned chain is confined to the negative half-line for $θ>1$, and the dynamics are genuinely two-sided and governed by $q$-trigonometric functions for $θ<0$. In each case, the persistence exponent, sharp persistence asymptotics, the defining eigenfunction, and the unique invariant distribution are obtained explicitly.

math.PR↗
arXiv2026

Persistence Probability of Fractional Brownian Motion with Random Hurst Exponent

We study the persistence properties of a fractional Brownian motion whose Hurst exponent is a random variable instead of a fixed constant. For each fixed $H \in (0,1)$, it is well known that the persistence probability of an FBM below a constant barrier decays like $T^{-(1-H)+o(1)}$, as $T$ tends to infinity, cf. Molchan (1999). Our object of interest is the persistence probability of the process resulting from first randomly selecting $H\in (0,1)$ and then considering a fractional Brownian motion with this value of $H$ as a Hurst exponent, a process that is referred to as a fractional Brownian motion with random exponent. We prove that its persistence probability decays as $T^{-(1-H_0)+o(1)}$, as $T$ tends to infinity, where $H_0$ is the essential supremum of the distribution of the random Hurst exponent.

math.PR↗
arXiv2026

Ornstein-Uhlenbeck process conditioned to have restricted $L_2$-norm

We condition an Ornstein-Uhlenbeck process on having an atypically small $L_2$-norm on long time intervals. The weak limit of these conditioned processes is again an Ornstein-Uhlenbeck process, this time with a stronger mean-reverting force than the unconditioned process, which is controlled by the restriction on the $L_2$-norm.

math.PR↗
arXiv2026

Persistence probabilities of fractional Lévy fields indexed by hyperbolic space and other Riemannian manifolds

We study the persistence probability of fractional Lévy fields, i.e. the analogue of fractional Brownian motion with generalised (multi-dimensional) index sets. First, we compute the persistence exponent of the hyperbolic fractional Lévy field. The result matches the rate obtained in Molchan (1999) for Euclidean space and the one in Aurzada/Helmer (2026) for the sphere. This enables us to study persistence for fractional Lévy fields indexed by a large class of Riemannian manifolds (whenever that process exists) through a local comparison argument with the spherical and hyperbolic case.

math.PR↗
arXiv2026

MA(1) processes with uniform innovations conditioned to stay positive in the non-expanding regime

We study an MA(1)-process with uniform innovations conditioned to stay positive. Representing the model as a Markov chain, we prove the existence of the limiting finite-dimensional distributions under this conditioning and identify the limiting process explicitly as a Doob $h$-transform. In the non-expanding case, i.e. when the coupling parameter $θ$ satisfies $θ\in[-1,1)$, we compute the relevant generating functions, extract sharp persistence asymptotics, and give explicit formulas for the eigenfunction $h$ and the persistence exponent. The resulting transition kernel of the limiting process is therefore fully explicit and displays a phase-dependent structure in the parameters. This provides a rare solvable example of a Markov chain on a continuous state space conditioned on persistence.

math.PR↗
arXiv2025

Persistence probabilities of MA(1) sequences with Laplace innovations and $q$-deformed zigzag numbers

We study the persistence probabilities of a moving average process of order one with innovations that follow a Laplace distribution. The persistence probabilities can be computed fully explicitly in terms of classical combinatorial quantities like certain $q$-Pochhammer symbols or $q$-deformed analogues of Euler's zigzag numbers, respectively. Similarly, the generating functions of the persistence probabilities can be written in terms of $q$-analogues of the exponential function or the $q$-sine/$q$-cosine functions, respectively.

math.PR↗
arXiv2025

Persistence probabilities for MA(1) sequences with uniform innovations

We study the persistence probabilities of a moving average process of order one with uniform innovations. We identify a number of regions, characterized by the location of the uniform distribution and the coupling parameter of the process, where the persistence probabilities have qualitatively different behaviour. We obtain the generating functions of the persistence probabilities explicitly in all possible regions. In some of the regions, the persistence probabilities can be expressed explicitly in terms of various combinatorial quantities.

math.PR↗
arXiv2025

The Best Time for an Update: Risk-Sensitive Minimization of Age-Based Metrics

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.

cs.IT↗
arXiv2025

Persistence probabilities of spherical fractional Brownian motion

We compute the rate of decay of the persistence probabilities of spherical fractional Brownian motion, which was defined by Lévy (1965) and Istas (2005). The rate resembles the Euclidean case treated in Molchan (1999). As a by-product we consider the coefficients of series representations of functions with algebraic endpoint singularities in terms of re-scaled Gegenbauer polynomials, which partly generalises Sidi (2009).

math.PR↗
arXiv2024

Persistence exponents via perturbation theory: MA(1)-processes

For the moving average process $X_n=ρξ_{n-1}+ξ_n$, $n\in\mathbb{N}$, where $ρ\in\mathbb{R}$ and $(ξ_i)_{i\ge -1}$ 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 $λ_ρ$ 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 $λ_ρ$ can be expressed as a power series in $ρ$. 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.

math.PR↗
arXiv2024

Occupation times and areas derived from random sampling

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 Lévy'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 Lévy processes.

math.PR↗
arXiv2024

Brownian motion conditioned to spend limited time outside a bounded interval -- an extreme example of entropic repulsion

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 $T \to \infty$.

math.PR↗
arXiv2024

Time- versus event-triggered consensus of a single-integrator multi-agent system

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.

eess.SY↗
arXiv2023

Persistence probabilities of a smooth self-similar anomalous diffusion process

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.

math.PR↗
arXiv2023

Scaling limit of stretched Brownian chains

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.

math.PR↗
arXiv2023

Persistence probabilities of weighted sums of stationary Gaussian sequences

With $\{ξ_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $ρ(i):=\mathbb{E}[ ξ_0ξ_i]$ and $\{σ(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell σ(i) ξ_i$, $\ell\ge 1$. For summable correlations $ρ$, we show that the persistence exponent is universal. On the contrary, for non-summable $ρ$, even for polynomial weight functions $σ(i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the polynomial rate $p$ of $σ$. In this case, we show existence of the persistence exponent $θ(H,p)$ 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.

math.PR↗
arXiv2022

Analysis of Time- versus Event-Triggered Consensus for a Single-Integrator Multi-Agent System

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.

eess.SY↗