Asymptotic Results Research Articles

Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion

We present an asymptotic result for the Laplace transform of the time integral of the geometric Brownian motion F(θ,T)=E[e−θXT] with XT=∫0TeσWs+(a−12σ2)sds, which is exact in the limit σ2T→0 at fixed σ2θT2 and aT. This asymptotic result is applied to pricing zero coupon bonds in the Dothan model of stochastic interest rates. The asymptotic result provides an approximation for bond prices which is in good agreement with numerical evaluations in a wide range of model parameters. As a side result we obtain the asymptotics for Asian option prices in the Black-Scholes model, taking into account interest rates and dividend yield contributions in the σ2T→0 limit.

Dual sourcing systems: An asymptotic result for the cost of optimal tailored base-surge (TBS) policy

We study the problem of managing an inventory system with two supply sources which differ in their lead times and unit prices. We consider a simple class of policies called Tailored Base-Surge (TBS) policies for managing this system. These policies order a constant amount in every period from the slow supplier and order up to a target level in every period from the faster and more expensive supplier. We prove an asymptotic convergence result on the cost of an optimal TBS policy as the unit inventory holding cost goes to zero while keeping the service level constant and simultaneously increasing the unit cost of procurement from the faster supplier. The asymptotic value of cost also serves as an upper bound on the optimal cost of all dual sourcing systems with the same service level and product of unit procurement and holding cost. This bound is better than the previously best known upper bound on the cost of an optimal dual sourcing policy in the same regime.

Asymptotic Fermat for signature (4,2,p) over number fields

Let K be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation x4−y2=zp over K, assuming some deep but standard conjectures of the Langlands programme when K has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above 2. When modularity of elliptic curves over K is known, for example when K is real quadratic or the r-layer of the cyclotomic Z2-extension of Q, effective asymptotic results hold.

A refined asymptotic result of one-dimensional flux limited Keller–Segel models

In this paper, we consider a flux-limited Keller–Segel model derived in [16, 18] in a one-dimensional bounded domain and give a refined asymptotic result of the spiky steady state by using the Sturm oscillation theorem in a more meticulous way based on the existence result of spiky steady state in [4], showing a more accurate characterization of the cell aggregation phenomenon.

On mixed censored [formula omitted]-shock models

In this paper we introduce the mixed censored δ-shock model which combines the censored δ-shock model and the classical extreme shock model. Under the mixed censored δ-shock model, the system fails whenever no shock occurs within a δ-length time period from the last shock, or the magnitude of the shock is larger than another critical threshold γ>0. For the discrete-time case of occurrence of shocks, by assuming the dependence between intershock times and the corresponding magnitudes of shocks we derive the probability generating function (pgf) of the lifetime of the system, and a matrix-based expression is obtained for the exact distribution of the system’s lifetime when the distribution of intershock times and the magnitudes of shocks have a discrete bivariate phase-type distribution. Similar results are obtained by assuming the independence between intershock times and the corresponding magnitudes of shocks, and by proving that the distribution of the shifted lifetime at δ, is the convolution of a discrete compound geometric distribution and a discrete compound Bernoulli distribution, we get several results concerning the distribution of system’s lifetime, like as, simple and efficient recursions for evaluating the survival function and the probability mass function (pmf), the mean and the variance of system’s lifetime, as well as discrete Lundberg-type upper bounds for the reliability function. For the continuous-time case of occurrence of shocks we obtain an exact formula for the reliability function and the Laplace–Stieltjes transform of system’s lifetime by assuming the dependence between intershock times and the corresponding magnitudes of shocks, and under the independence setup, we obtain the reliability function when the intershock times have the uniform distribution, and we give an asymptotic result under the Poisson process for the arrival of shocks. Similar to the discrete-time case, it is shown that the distribution of the shifted lifetime at δ, is the convolution of a compound geometric distribution and a compound Bernoulli distribution and using this we obtain two-sided Lundberg-type bounds for the survival function. Finally, some numerical examples to illustrate our results, are also given.

Infinite ergodicity in generalized geometric Brownian motions with nonlinear drift.

Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g., in finance, in physics, and biology. The definition of the process depends crucially on the interpretation of the stochastic integrals which involves the discretization parameter α with 0≤α≤1, giving rise to the well-known special cases α=0 (Itô), α=1/2 (Fisk-Stratonovich), and α=1 (Hänggi-Klimontovich or anti-Itô). In this paper we study the asymptotic limits of the probability distribution functions of geometric Brownian motion and some related generalizations. We establish the conditions for the existence of normalizable asymptotic distributions depending on the discretization parameter α. Using the infinite ergodicity approach, recently applied to stochastic processes with multiplicative noise by E. Barkai and collaborators, we show how meaningful asymptotic results can be formulated in a transparent way.

Large Deviations for Hawkes Processes with Randomized Baseline Intensity

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics.

Shaping effects on the geodesic acoustic mode in tokamaks

The geodesic acoustic mode (GAM) is investigated with gyro-kinetic equations in Miller local equilibrium model for shaped tokamak plasmas with an arbitrary elongation κ and a finite triangularity δ. In particular, the effects of triangularity on GAM frequency and damping rate are analyzed both analytically and numerically. The asymptotic analytical and exact numerical results both show that the frequency almost linearly increases with the triangularity but increases relatively more slowly for a negative δ, which agree well with the TCV observation on the trend. The analytical results clearly claim that the triangularity effect strength is dependent on the inverse aspect ratio and Shafranov shift gradient , while the numerical results indicate that the safety factor q also has a significant impact on the triangularity effects. In addition, the damping rate increases rapidly with triangularity when q is not too large and then saturates when δ is above about 0.3.

Front propagation in the shadow wave-pinning model.

In this paper we consider a non-local bistable reaction-diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x,t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain [Formula: see text], separated by an interface moving at a normal velocity regulated by the integral [Formula: see text]. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity [Formula: see text], we prove that for any [Formula: see text] the interface can be estimated within [Formula: see text] of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

Asymptotic results of a multiple-entry reinforcement process

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions {Pt}t≥1, where Pt is a partition of {1,2,…,Rt}. At each time t, R numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block’s cardinality, independently. We prove results concerning the asymptotic cardinality of a given block and central limit theorems for associated fluctuations about this asymptotic cardinality: these are proved both for a fixed block and for the maximum among all blocks. We also prove that with probability one, a single block eventually takes and maintains the leadership in cardinality. Depending on the way one sees this partition process, one can translate our results to Balls and Bins processes, Generalized Chinese Restaurant Processes, Generalized Urn models and Preferential attachment random graphs.

Event-Based Output Feedback Consensus Control for Multiagent Systems With Unknown Non-Identical Control Directions

This article studies the event-triggered output feedback consensus problem for a class of high-order uncertain nonlinear multiagent systems with totally unknown non-identical control directions. Firstly, the reduced-order filters are constructed by utilizing the dynamic gain technique to compensate the system uncertainties and unmeasurable states of the followers. Based on the filter states and output information, the asymptotic output consensus result is guaranteed with the developed distributed adaptive output feedback control protocol, while the inherent issue “complexity explosion” in backstepping design is avoided. Then, to solve the interaction of multiple Nussbaum functions in a single inequality, the Nussbaum functions with different frequencies are adopted, with which the unknown directions of multiagent system are no longer limited in identical. In addition, we propose a dynamic triggering strategy with time-varying threshold parameters to reduce the updates of controller without Zeno phenomenon. Finally, simulation results are provided to illustrate the effectiveness of theoretical algorithm.

Event-Triggered Input–Output Finite-Time Stabilization for IT2 Fuzzy Systems Under Deception Attacks

This article addresses the issue of input–output finite-time stabilization for interval type-2 fuzzy systems in the presence of deception attack effects. Our main goal is to make efficient use of network resources by developing an event-triggered controller for interval type-2 fuzzy systems. For the stabilization process, an event-triggered controller that does not share the same membership functions as the system is designed by using affine transformation parameters. Following that, by using an asymmetric Lyapunov–Krasovskii functional and some advanced integral inequalities to establish the sufficient conditions for the existence of the proposed controller. Furthermore, an asymptotic stabilization result is presented and discussed in a comparative analysis as a special case. Due to the asymmetric Lyapunov–Krasovskii functional structure, the transmission delay interval is large compared with some recent studies. Finally, two simulation examples are carried out and the efficiency of designed controller is verified.

New asymptotic results for generalized Oppenheim expansions

New asymptotic results for generalized Oppenheim expansions

(Non)-Coherent MU-MIMO Block Fading Channels With Finite Blocklength and Linear Processing

Driven by the stringent demands of future ultra reliable and low latency communication (URLLC), we provide a comprehensive study for a coherent and non-coherent multiuser multiple-input multiple-output (MU-MIMO) uplink system in the finite blocklength regime. The independent and identically distributed (i.i.d.) Gaussian codebook is assumed for each user. To be more specific, the base station (BS) first uses two popular linear processing schemes to combine the signals transmitted from all users, namely maximum-ratio combining (MRC) and zero-forcing (ZF). Following it, the matched maximum-likelihood (ML) and mismatched nearest-neighbour (NN) decoding metric for the coherent and non-coherent cases are respectively employed at the BS. Under these conditions, the refined third-order achievable coding rate, expressed as a function of the blocklength, average error probability, and the third-order term of the information density (called as the channel perturbation), is derived. With this result in hand, a detailed performance analysis is then pursued, through which, we derive the asymptotic results of the channel perturbation, achievable coding rate, channel capacity, and the channel dispersion. These theoretical results enable us to obtain a number of interesting insights related to the impact of the finite blocklength: i) in our system setting, massive MIMO helps to reduce the channel perturbation of the achievable coding rate, which can even be discarded without affecting the performance with just a small-to-moderate number of BS antennas and number of blocks; ii) under the non-coherent case, even with massive MIMO, the channel estimation errors cannot be eliminated unless the transmit powers in both the channel estimation and data transmission phases for each user are made inversely proportional to the square root of the number of BS antennas; iii) in the non-coherent case and for fixed total blocklength, the scenarios with longer coherence intervals and smaller number of blocks will offer higher achievable coding rate.

Model identification via total Frobenius norm of multivariate spectra

Abstract We study the integral of the Frobenius norm as a measure of the discrepancy between two multivariate spectra. Such a measure can be used to fit time series models, and ensures proximity between model and process at all frequencies of the spectral density. We develop new asymptotic results for linear and quadratic functionals of the periodogram, and apply the integrated Frobenius norm to fit time series models and test whether model residuals are white noise. The case of structural time series models is addressed, wherein co-integration rank testing is formally developed. Both applications are studied through simulation studies and time series data. The numerical results show that the proposed estimator can fit moderate- to large-dimensional structural timeseries in real time.

THE GROWTH OF SOLUTIONS OF MONGE–AMPÈRE EQUATIONS IN HALF SPACES AND ITS APPLICATION

AbstractWe consider the growth of the convex viscosity solution of the Monge–Ampère equation $\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary $\{x_n=0\}$ and there exists some $\varepsilon>0$ such that $u=O(|x|^{3-\varepsilon })$ at infinity, then $u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations269(1) (2020), 326–348].

Oscillatory translational instabilities of spot patterns in the Schnakenberg system on general 2D domains

For a bounded 2D planar domain Ω, we investigate the impact of domain geometry on oscillatory translational instabilities of N-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An N-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in Ω except in N well-separated localized regions of extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in -frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a 2N-vector that characterizes the possible modes (directions) of oscillation. The matrix contains terms associated with a certain Green’s function on Ω, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates with , , and 2π-periodic, we show that only the mode-2 coefficients of the Fourier series of f impact the bifurcation threshold at leading order in σ. We further show that when , the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains Ω and N-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.

Triple Sampling Inference Procedures for the Mean of the Normal Distribution When the Population Coefficient of Variation Is Known

This paper discusses the triple sampling inference procedures for the mean of a symmetric distribution—the normal distribution when the coefficient of variation is known. We use the Searls’ estimator as an initial estimate for the unknown population mean rather than the classical sample mean. In statistics literature, the normal distribution under investigation underlines almost all the natural phenomena with applications in many fields. First, we discuss the minimum risk point estimation problem under a squared error loss function with linear sampling cost. We obtained all asymptotic results that enhanced finding the second-order asymptotic risk and regret. Second, we construct a fixed-width confidence interval for the mean that satisfies at least a predetermined nominal value and find the second-order asymptotic coverage probability. Both estimation problems are performed under a unified optimal framework. The theoretical results reveal that the performance of the triple sampling procedure depends on the numerical value of the coefficient of variation—the smaller the coefficient of variation, the better the performance of the procedure.

Outage Performance and AoI Minimization of HARQ-IR-RIS Aided IoT Networks

In this paper, reconfigurable intelligent surface (RIS) and hybrid automatic repeat request with incremental redundancy (HARQ-IR) are amalgamated to lower power expenditure, shorten latency and strengthen reliability of the internet of things (IoT) communications. By considering Rician fading channels and multiple RISs, the outage probability of single-input single-output (SISO) HARQ-IR-RIS aided IoT networks is derived in closed-form, with which the asymptotic outage analysis is carried out. The asymptotic results are further extended to single-input multiple-output (SIMO)/multiple-input multiple-output (MIMO) HARQ-IR-RIS aided IoT networks by using random matrix theory. Thereafter, the asymptotic expressions are invoked to reduce the design complexity of the phase shifts, transmit powers, and rate, which aims at minimizing the age of information (AoI) while ensuring the power and outage constraints. Particularly, the optimal phase shifts are firstly determined. The alternating optimization technique is then applied to solve the transmit rate and powers, which are iteratively updated based on majorization-minimization (MM) principle and geometric programming (GP) approximation, respectively. The numerical results consequently corroborate our theoretical analysis. More interestingly, the HARQ-IR-aided scheme with fixed power is found to provide a comparable performance as the proposed scheme with variable power especially for numerous reflecting elements at RIS.

Nonlinear three-dimensional patterns of the Marangoni convection in a thin film on a poorly conducting substrate.

We investigate the dynamics of a thin liquid film that is placed atop a heated substrate of very low thermal conductivity. The direct numerical simulation of the stationary long-wave Marangoni instability is performed with the system of coupled partial differential equations. These equations were previously derived within the lubrication approximation; they describe the evolution of film thickness and fluid temperature. We compare our results with the early reported results of the weakly nonlinear analysis. A good qualitative agreement is observed for values of the Marangoni number near the convective threshold. In the case of supercritical excitation, our results for the amplitudes are described by the square root dependence on the supercriticality. In the case of subcritical excitation, we report the hysteresis. For relatively high supercriticality, the convective regimes evolve into film rupture via the emergence of secondary humps. For the three-dimensional patterns, we observe rolls or squares, depending on the problem parameters. We also confirm the prediction of the asymptotic results concerning the nonlinear feedback control for the pattern selection. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.