First-order Asymptotics Research Articles

Networks of reinforced stochastic processes: A complete description of the first-order asymptotics

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions for the emergence of some form of almost sure asymptotic synchronization. Specifically, we identify three regimes: the first involves complete synchronization, where all processes converge towards the same random variable; the second exhibits almost sure convergence of the system, but no form of synchronization subsists; and the third reveals a scenario where there is almost sure asymptotic synchronization within the cyclic classes of the interaction matrix, together with an asymptotic periodic behavior among these classes.

Asymptotics of Cointegration Tests for High-Dimensional Var(k)

Abstract The paper studies nonstationary high-dimensional vector autoregressions of order k, VAR(k). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, T, and the number of coordinates, N, are assumed to be large and of the same order. Under this regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai–Bartlett, and Hotelling–Lawley tests for cointegration are derived: the test statistics converge to nonrandom integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy1 point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond those considered in our theorems. The paper presents empirical implementations of the approach for the analysis of S&P100 stocks and of cryptocurrencies. The latter example has a strong presence of multiple cointegrating relationships, while the results for the former are consistent with the null of no cointegration.

Asymptotics for short maturity Asian options in jump-diffusion models with local volatility

We present a study of the short maturity asymptotics for Asian options in a jump-diffusion model with a local volatility component, where the jumps are modeled as a compound Poisson process. The analysis for out-of-the-money Asian options is extended to models with Lévy jumps, including the exponential Lévy model as a special case. Both fixed and floating strike Asian options are considered. Explicit results are obtained for the first-order asymptotics of the Asian options prices for a few popular models in the literature: the Merton jump-diffusion model, the double-exponential jump model, and the Variance Gamma model. We propose an analytical approximation for Asian option prices which satisfies the constraints from the short-maturity asymptotics, and test it against Monte Carlo simulations. The asymptotic results are in good agreement with numerical simulations for sufficiently small maturity.

First-order heat content asymptotics on [formula omitted] spaces

In this paper, we prove first-order asymptotics on a bounded open set of the heat content when the ambient space is an RCD(K,N) space, under a regularity condition for the boundary that we call measured interior geodesic condition of size ϵ. We carefully study such a condition, relating it to the properties of the disintegration of the signed distance function from ∂Ω studied in Cavalletti and Mondino(2020).

Uncertainty quantification in the Bradley–Terry–Luce model

Abstract The Bradley–Terry–Luce (BTL) model is a benchmark model for pairwise comparisons between individuals. Despite recent progress on the first-order asymptotics of several popular procedures, the understanding of uncertainty quantification in the BTL model remains largely incomplete, especially when the underlying comparison graph is sparse. In this paper, we fill this gap by focusing on two estimators that have received much recent attention: the maximum likelihood estimator (MLE) and the spectral estimator. Using a unified proof strategy, we derive sharp and uniform non-asymptotic expansions for both estimators in the sparsest possible regime (up to some poly-logarithmic factors) of the underlying comparison graph. These expansions allow us to obtain: (i) finite-dimensional central limit theorems for both estimators; (ii) construction of confidence intervals for individual ranks; (iii) optimal constant of $\ell _2$ estimation, which is achieved by the MLE but not by the spectral estimator. Our proof is based on a self-consistent equation of the second-order remainder vector and a novel leave-two-out analysis.

Counting phylogenetic networks with few reticulation vertices: A second approach

Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number TC ℓ , k of tree-child networks with ℓ leaves and k reticulation vertices has the first-order asymptotics TC ℓ , k ∼ c k 2 e ℓ ℓ ℓ + 2 k − 1 , ( ℓ → ∞ ) . Moreover, they also computed c k for k = 1 , 2 , and 3. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for c k , namely, c k = 2 k − 1 2 / k ! for all k ≥ 0 .

Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations

AbstractLet $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

Dispersion for Schrödinger operators on regular trees

We prove dispersive estimates for two models: the adjacency matrix on a discrete regular tree, and the Schrödinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schrödinger operators on the real line. We establish a \(t^{-3/2}\)-decay for both models which is sharp, as we give the first-order asymptotics.

Critical value asymptotics for the contact process on random graphs

Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.

Asymptotic enumeration and distributional properties of galled networks

We show a first-order asymptotics result for the number of galled networks with n leaves. This is the first class of phylogenetic networks of large size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of galled networks with n leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) [12] which was devised for the purpose of (exactly) counting galled networks. Moreover, an old result of Bender and Richmond (1984) [1] plays a crucial role in our proofs, too.

Saddlepoint Approximations for Spatial Panel Data Models

We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail area approximation feature relative error of order with n being the cross-sectional dimension and T the time-series dimension. The main theoretical tool is the tilted-Edgeworth technique in a nonidentically distributed setting. The density approximation is always nonnegative, does not need resampling, and is accurate in the tails. Monte Carlo experiments on density approximation and testing in the presence of nuisance parameters illustrate the good performance of our approximation over first-order asymptotics and Edgeworth expansion. An empirical application to the investment–saving relationship in OECD (Organisation for Economic Co-operation and Development) countries shows disagreement between testing results based on the first-order asymptotics and saddlepoint techniques. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and H\'elein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress-energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.

Asymptotics of solutions to the problem of fluid outflow from a rectangular duct

We investigated the asymptotics of two-dimensional steady solutions simulating the energy-conserving flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the fluid surface detaches from the ceiling of the duct as a free surface. These asymptotics are constructed using the local hydrostatic approximation, which generalizes the classical long-wave approximation. The initial (zero-order) asymptotics leading to the piecewise constant solutions are obtained from the mass, momentum, and energy conservation laws of the first approximation of shallow water theory. The first-order asymptotics for the liquid depth are constructed using the momentum conservation law of the Green–Nagdi model representing the second approximation of shallow water theory. It is shown that the continuous solution obtained from this asymptotics is in good agreement with the Wilkinson laboratory experiment [D. L. Wilkinson, “Motion of air cavities in long horizontal ducts,” J. Fluid Mech. 118, 109 (1982)] on modeling the energy-conserving steady flow predicted by the classical piecewise constant Benjamin solution [T. B. Benjamin, “Gravity currents and related phenomena,” J. Fluid Mech. 31, 209 (1968)].

Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

First and Second Order Asymptotics of the Spectral Risk Measure for Portfolio Loss Under Multivariate Regular Variation

In the context of multivariate regular variation, the authors establish the first-order asymptotics of the spectral risk measure of portfolio loss. Furthermore, by the notion of second-order regular variation, the second-order asymptotics of the spectral risk measure of portfolio loss is also presented. In order to illustrate the derived results, a numerical example with Monte Carlo simulation is carried out.

On Binomial and Poisson Sums Arising from the Displacement of Randomly Placed Sensors

We re-visit the asymptotics of a binomial and a Poisson sum which arose as (average) displacement costs when moving randomly placed sensors to anchor positions. The first-order asymptotics of these sums were derived in several stages in a series of recent papers. In this paper, we give a unified approach based on the classical Laplace method with which one can also derive more terms in the asymptotic expansions. Moreover, in a special case, full asymptotic expansions can be given which even hold as identities. This will be proved by a combinatorial approach and systematic ways of computing all coefficients of these identities will be discussed as well.

Sequential minimum risk point estimation (MRPE) methodology for a normal mean under Linex loss plus sampling cost: First-order and second-order asymptotics

We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes.

Local probabilities of randomly stopped sums of power-law lattice random variables

Let X1 and N ≥ 0 be integer-valued power-law random variables. For a randomly stopped sum SN = X1+⋯+XN of independent and identically distributed copies of X1, we establish a first-order asymptotics of the local probabilities P(SN = t) as t → +1 ∞. Using this result, we show the scaling k-δ, 0 ≤ δ ≤ 1, of the local clustering coefficient (of a randomly selected vertex of degree k) in a power-law affiliation network.

Saddlepoint Approximations for Spatial Panel Data Models

We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail area approximation feature relative error of order $O(1/(n(T-1)))$ with $n$ being the cross-sectional dimension and $T$ the time-series dimension. The main theoretical tool is the tilted-Edgeworth technique in a non-identically distributed setting. The density approximation is always non-negative, does not need resampling, and is accurate in the tails. Monte Carlo experiments on density approximation and testing in the presence of nuisance parameters illustrate the good performance of our approximation over first-order asymptotics and Edgeworth expansions. An empirical application to the investment-saving relationship in OECD (Organisation for Economic Co-operation and Development) countries shows disagreement between testing results based on first-order asymptotics and saddlepoint techniques.

First- and Second-Order Asymptotics in Covert Communication

We study the first- and second-order asymptotics of covert communication over binary-input DMC for three different covertness metrics and under maximum probability of error constraint. When covertness is measured in terms of the relative entropy between the channel output distributions induced with and without communication, we characterize the exact first- and second-order asymptotics of the number of bits that can be reliably transmitted with a maximum probability of error less than $\epsilon$ and a relative entropy less than $\delta$. When covertness is measured in terms of the variational distance between the channel output distributions or in terms of the probability of missed detection for fixed probability of false alarm, we establish the exact first-order asymptotics and bound the second-order asymptotics. PPM achieves the optimal first-order asymptotics for all three metrics, as well as the optimal second-order asymptotics for relative entropy. The main conceptual contribution of this paper is to clarify how the choice of a covertness metric impacts the information-theoretic limits of covert communications. The main technical contribution underlying our results is a detailed expurgation argument to show the existence of a code satisfying the reliability and covertness criteria.