Arcsine Law Research Articles

Arcsine Laws of Light.

We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.

One-bit spectrum sensing using Gustafson–Kessel fuzzy clustering for cognitive radio network

In this article, we address the problem of one-bit spectrum sensing using a multi-antenna array for cognitive radio networks. Spectrum sensing with one-bit analog-to-digital converter (ADC) is a far-reaching technology as it requires only a simple comparator for conversion and has the advantage of low-cost and high-speed. However, spectrum sensing with one-bit quantized data is challenging because of severe loss of information due to nonlinearities involved. Existing one-bit spectrum sensing techniques utilize arcsine law to approximate original signal covariance matrix from their quantized version and then analytically model the signal features for deriving the spectrum classification rule. However, these techniques are vulnerable to the inaccuracies of the model and it may not be at all possible to derive a classification rule for analytically intractable noise scenarios. In this article, we follow a data-driven approach for one-bit spectrum sensing and propose Gustafson–Kessel fuzzy c-means (GKFCM) clustering based algorithms for detection of primary user signals in white and correlated noise. We identify the features which are conserved in the process of quantization and form decision vectors using them. Decision vectors obtained from the received one-bit quantized data are then clustered using GKFCM to obtain the model parameters and classification is performed to identify the occupancy of the channel. Simulation results reveal that proposed algorithms provide significant improvement in detection performance compared to existing one-bit spectrum sensing techniques in literature.

Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

In this work, we investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant D) during a time interval 0,t in the presence of a reflective boundary at the origin, when starting from a positive position x0. We first obtain the distribution P(M|x0,t) of the maximum displacement M and its expectation 〈M〉. In the short-time limit, i.e., t≪td where td=x02/D is the diffusion time from the starting position x0 to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, t≫td, 〈M〉 grows with t as 〈M〉∼t, which is similar to the free Brownian motion, but the prefactor is π/2 times of the free Brownian motion, embodying the effect of the reflective boundary. By solving the propagator and using a path decomposition technique, we then obtain the joint distribution P(M,tm|x0,t) of M and the time tm at which this maximum is achieved, from which the marginal distribution P(tm|x0,t) of tm is also obtained. For t≪td, P(tm|x0,t) looks like a U-shaped attributed to the arcsine law of free Brownian motion. For t equal to or larger than order of magnitude of tm, P(tm|x0,t) deviates from the U-shaped distribution and becomes asymmetric with respect to t/2. Moreover, we compute the expectation 〈tm〉 of tm, and find that 〈tm〉/t is an increasing function of t. In two limiting cases, 〈tm〉/t→1/2 for t≪td and 〈tm〉/t→(1+2G)/4≈0.708 for t≫td, where G≈0.916 is the Catalan’s constant. Finally, we analytically compute the statistics of the last time tℓ the particle crosses the starting position x0 and the occupation time to spent above x0. We find that 〈tℓ〉/t→1/2 in the short-time and long-time limits, and reaches its maximum at an intermediate value of t. The fraction of the occupation time 〈to〉/t is a monotonic function of t, and tends towards 1 in the long-time limit. All the theoretical results are validated by numerical simulations.

Breakdown of arcsine law for resetting brownian motion

For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time V staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position x r at random times but with a constant rate r. When x r is exactly equal to zero, we derive the exact expression of the probability distribution P r (V∣0, t) of V during time t, and the moments of V as functions of r and t. P r (V∣0, t) is always symmetric with respect to V = t/2 for arbitrary value of r, but the probability density of V at V = t/2 increases with the increase of r. Interestingly, P r (V∣0, t) at V = t/2 changes from a minimum to a local maximum at a critical value R * ≈ 0.742 338, where R = rt denotes the average number of resetting during time t. Moreover, we consider the case when x r is a random variable and is distributed by a function g(x r ), where g(x r ) is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of V when the variance of x r is low. The mean value of V is always equal to t/2, but the fluctuation in x r leads to an increase in the second and third moments of V. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.

Statistics of the number of renewals, occupation times, and correlation in ordinary, equilibrium, and aging alternating renewal processes.

The renewal process is a point process where an interevent time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the interevent time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the interevent times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the interevent-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of interevent time diverges. When the mean interevent time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.

Extreme value statistics and Arcsine laws of Brownian motion in the presence of a permeable barrier

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follow the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman–Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form.

Convergence Rates to the Arcsine Law

Convergence Rates to the Arcsine Law

Records and Occupation Time Statistics for Area-Preserving Maps

A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated with stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records statistics, and occupation time statistics. Our results confirm and expand known results for the standard map: when a chaotic sea is present, transport is diffusive, and records statistics and the fraction of occupation time in the positive half-axis reproduce the laws for simple symmetric random walks. In the case of the triangle map, we retrieve the previously observed anomalous transport, and we show that records statistics exhibit similar anomalies. When we investigate occupation time statistics and persistence probabilities, our numerical experiments are compatible with a generalized arcsine law and transient behavior of the dynamics.

Arcsine law for random dynamics with a core

In their recent paper, Hata and Yano (2021 Stoch. Dyn. 2350006) first gave an example of random iterations of two piecewise linear interval maps without (deterministic) indifferent periodic points for which the arcsine law—a characterization of intermittent dynamics in infinite ergodic theory—holds. The key in the proof of the result is the existence of a Markov partition preserved by each interval maps. In the present paper, we give a class of random iterations of two interval maps without indifferent periodic points but satisfying the arcsine law, by introducing a concept of core random dynamics. As applications, we show that the generalized arcsine law holds for generalized Hata–Yano maps and piecewise linear versions of Gharaei–Homburg maps, both of which do not have a Markov partition in general.

Covariance recovery for one-bit sampled stationary signals with time-varying sampling thresholds

One-bit quantization, which relies on comparing the signals of interest with given threshold levels, has attracted considerable attention in signal processing for communications and sensing. A useful tool for covariance recovery in such settings is the arcsine law, that estimates the normalized covariance matrix of zero-mean stationary input signals. This relation, however, only considers a zero sampling threshold, which can cause a remarkable information loss. In this paper, the idea of the arcsine law is extended to the case where one-bit analog-to-digital converters (ADCs) apply time-varying thresholds. Specifically, three distinct approaches are proposed, investigated, and compared, to recover the autocorrelation sequence of the stationary signals of interest. Additionally, we will study a modification of the Bussgang law, a famous relation facilitating the recovery of the cross-correlation between the one-bit sampled data and the zero-mean stationary input signal. Similar to the case of the arcsine law, the Bussgang law only considers a zero sampling threshold. This relation is also extended to accommodate the more general case of time-varying thresholds for the stationary input signals.

The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the kth moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the \(p^m\)th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

Experimental verification of arcsine laws in mesoscopic nonequilibrium systems.

A large number of processes in the mesoscopic world occur out of equilibrium, where the time evolution of a system becomes immensely important since it is driven principally by dissipative effects. Nonequilibrium steady states (NESS) represent a crucial category in such systems, where relaxation timescales are comparable to the operational timescales. In this study, we employ a model NESS stochastic system, which is comprised of a colloidal microparticle optically trapped in a viscous fluid, externally driven by a temporally correlated noise, and show that time-integrated observables such as the entropic current, the work done on the system or the work dissipated by it, follow the three Lévy arcsine laws [A.C. Barato et al., Phys. Rev. Lett. 121, 090601 (2018)0031-900710.1103/PhysRevLett.121.090601], in the large time limit. We discover that cumulative distributions converge faster to arcsine distributions when it is near equilibrium and the rate of entropy production is small, because in that case the entropic current has weaker temporal autocorrelation. We study this phenomenon by changing the strength of the added noise as well as by perturbing our system with a flow field produced by a microbubble at close proximity to the trapped particle. We confirm our experimental findings with theoretical simulations of the systems. Our work provides an interesting insight into the NESS statistics of the meso-regime, where stochastic fluctuations play a pivotal role.

Rate of Convergence, Relation Between the Arcsine Law and P(τ≤n) in the Optimal Stopping Problem For S_n/n

We establish the relation between the arcsine law and P(τ ≤n), and we prove that 〖Sup〗_(t≤N) E S_t/t→ 〖Sup〗_(t≤∞) E S_t/t as N →∞ and the rate of convergence belongs between 1/(2(N+1)) E |X_1 | ∀ N and (1+ε ) 2/N^(3/4) for arbitrary ε> 0, N>N_ε .

Extreme value statistics and arcsine laws for heterogeneous diffusion processes.

Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

Covariance Recovery for One-Bit Sampled Non-Stationary Signals With Time-Varying Sampling Thresholds

The recovery of the input signal covariance values from its one-bit sampled counterpart has been deemed a challenging task in the literature. To deal with its difficulties, some assumptions are typically made to find a relation between the input covariance matrix and the autocorrelation values of the one-bit sampled data. This includes the arcsine law and the recently proposed modified arcsine law which unleashes a promising performance in covariance recovery by taking advantage of time-varying sampling thresholds. However, the modified arcsine law also assumes input signals are stationary, which is typically a simplifying assumption for real-world applications. In fact, in many signal processing applications, the input signals are readily known to be non-stationary with a non-Toeplitz covariance matrix. In this paper, we propose an approach to extending the arcsine law to the case where one-bit ADCs apply time-varying thresholds while dealing with input signals that originate from a non-stationary process. In particular, the recovery methods are shown to accurately recover the time-varying variance and autocorrelation values. Furthermore, we extend the formulation of the Bussgang law to the case where non-stationary input signals are considered.

A note on Brownian areas and arcsine laws

Firstly, we provide simple elementary proofs to derive the exact distributions of the areas under functions of a Brownian motion process and a Brownian bridge process. In the latter case, a solution is therefore provided to a question raised recently in the Mathematics community on StackExchange (http://math.stackexchange.com/questions/1006101). These random areas often occur in statistical applications and play an important role in, for example, financial mathematics. Comparisons are made between the variances of the two random areas, deriving interesting results that appear to be new in the statistical literature. Some illustrative examples are provided. Secondly, we derive a new arcsine law for a standard Brownian bridge process. Keywords: Arcsine law, Brownian bridge, Brownian motion, Gaussian process, Itô stochastic calculus

Determination of probabilistic characteristics of random values of estimates of the Lyapunov function when describing a physical process

The applied application of the Lyapunov characteristic function is determined by the properties of its estimates. Probabilistic characteristics of estimates of the Lyapunov characteristic function are described for the first time. The probabilistic characteristics of random values of estimates of the Lyapunov function are empirically estimated using statistical methods. The Matlab package has developed a model of a special device for obtaining estimates of the characteristic function by a direct method. A quasi-deterministic signal is fed to the input of the model, the instantaneous values of which are distributed according to the arcsine law, and an array of values of estimates of the Lyapunov function is obtained at the output, which is used to estimate the probabilistic characteristics of these estimates. Statistical estimation was performed by an indirect method. It is established that the values of the estimates of the Lyapunov characteristic function are distributed according to the normal law. The results of the research will be useful in engineering calculations, for example, when detecting message transmission errors in modems with a modulated characteristic function.

Arcsine laws for random walks generated from random permutations with applications to genomics

AbstractA classical result for the simple symmetric random walk with 2nsteps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows(q) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step 2n for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Stein’s method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows(q) permutation which is of independent interest.

Functionals of fractional Brownian motion and the three arcsine laws.

Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent H∈(0,1), generalizing standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for nonequilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many nontrivial observables analytically: We generalize the celebrated three arcsine laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in ɛ=H-1/2, up to second order. We find that the three probabilities are different, except for H=1/2, where they coincide. Our results are confirmed to high precision by numerical simulations.

Arcsine law and multistable Brownian dynamics in a tilted periodic potential.

Multistability is one of the most important phenomena in dynamical systems, e.g., bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures, to name only a few. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to the arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices, and colloidal systems.