Exact Asymptotics Research Articles

Stationary distributions for two-dimensional sticky Brownian motions: Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. In this paper, we focus on stationary distributions for sticky Brownian motions. Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions. The kernel method, copula concept and extreme value theory are the main tools used in our analysis.

Asymptotics for Capelli polynomials with involution

Let be the free associative algebra with involution * over a field F of characteristic zero. We study the asymptotic behavior of the sequence of *-codimensions of the T-*-ideal of generated by the *-Capelli polynomials and alternanting on M + 1 symmetric variables and L + 1 skew variables, respectively. It is well known that, if F is an algebraic closed field of characteristic zero, every finite dimensional *-simple algebra is isomorphic to one of the following algebras: the algebra of k × k matrices with the transpose involution; the algebra of matrices with the symplectic involution; the direct sum of the algebra of h × h matrices and the opposite algebra with the exchange involution. We prove that the *-codimensions of a finite dimensional *-simple algebra are asymptotically equal to the *-codimensions of for some fixed natural numbers M and L. In particular: and Moreover the exact asymptotics of and are known and those of can be easily deduced.

Sine-Gordon on a wormhole

In an attempt to understand the soliton resolution conjecture, we consider the sine-Gordon equation on a spherically symmetric wormhole spacetime. We show that within each topological sector (indexed by a positive integer degree n) there exists a unique linearly stable soliton, which we call the n-kink. We give numerical evidence that the n-kink is a global attractor in the evolution of any smooth, finite energy solutions of degree n. When the radius of the wormhole throat a is large enough, the convergence to the n-kink is shown to be governed by internal modes that slowly decay due to the resonant transfer of energy to radiation. We compute the exact asymptotics of this relaxation process for the one-kink using the Soffer–Weinstein weakly nonlinear perturbation theory.

Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant

We consider an obliquely reflected Brownian motion Z with positive drift in a quadrant stopped at time T, where T≔inf{t>0:Z(t)=(0,0)} is the first hitting time at the origin. Such a process can be defined even in the non-standard case in which the reflection matrix is not completely-S. We show in this case that the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in finite time. Given an arbitrary starting point (u,v) in the quadrant, we consider the escape (resp. absorption) probabilities P(u,v)[T=∞] (resp. P(u,v)[T<∞]). We establish the partial differential equations and the oblique Neumann boundary conditions which characterize the escape probability and provide a functional equation satisfied by the Laplace transform of the escape probability. Asymptotics for the absorption probability in the simpler case in which the starting point in the quadrant is (u,0) are then given. We proceed to show a geometric criterion on the parameters which characterizes the case in which the absorption probability has a product form and is exponential. We call this new criterion the dual skew symmetry condition due to its natural connection with the skew symmetry condition for the stationary distribution. We then obtain an explicit integral expression for the Laplace transform of the escape probability and conclude by presenting exact asymptotics for the escape probability at the origin.

The number of $D_4$-fields ordered by conductor

We consider families of quartic number fields whose normal closures over \mathbb{Q} have Galois group isomorphic to D_4 , the symmetries of a square. To any such field L , one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D_4 . We determine the asymptotic number of such D_4 -quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L -function methods. Both of these strategies fail in the case of D_4 -quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D_4 combined with both approaches mentioned above to obtain exact asymptotics.

Local theorems for (multidimensional) additive functionals of semi-Markov chains

We consider a (multidimensional) additive functional of semi-Markov chain, defined by an ergodic Markov chain with a finite number of states. The distribution of random vectors, governing the process, is supposed to be lattice and light-tailed. We derive the exact asymptotics in the local limit theorem. As a consequence, we establish a local central limit theorem.

Ruin Probability for a Gaussian Process with Variance Attaining its Maximum on Discrete Sets

Ruin probability for a Gaussian locally stationary process is considered in the case where the process variance attains its maximum in a finite number of points. The double sum method is applied to calculate exact asymptotics of the corresponding probability. Also, we consider a family of processes with variance that has a countable set of maximum points containing a limit point.

Some Properties of the Kilbas-Saigo Function

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

On Exact Asymptotics of the Error Probability in Channel Coding: Symmetric Channels

The exact order of the optimal sub-exponentially decaying factor in the classical bounds on the error probability of fixed-length codes over a Gallager-symmetric discrete memoryless channel with and without ideal feedback is determined for rates above the critical rate. Regardless of the availability of feedback, it is shown that the order of the optimal sub-exponential factor exhibits a dichotomy. Moreover, the proof technique is used to establish the third-order term in the normal approximation for symmetric channels, where a similar dichotomy is shown to exist.

Approximate Voronoi cells for lattices, revisited

Abstract We revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis–Laarhoven–De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than 2 0.076 d + o ( d ) $2^{0.076d + o(d)}$ memory, and we show how to obtain time–memory trade-offs even when using less than 2 0.048 d + o ( d ) $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as d d / 2 + o ( d ) $d^{d/2 + o(d)}$ matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

Exact Asymptotics for Learning Tree-Structured Graphical Models With Side Information: Noiseless and Noisy Samples

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan et al. (2011) and strictly improve those of Bresler and Karzand (2020). In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis et al. (2019). Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

Stationary Waiting Time in Parallel Queues with Synchronization

Motivated by database locking problems in today’s massive computing systems, we analyze a queueing network with many servers in parallel (files) to which jobs (writing access requests) arrive according to a Poisson process. Each job requests simultaneous access to a random number of files in the database and will lock them for a random period of time. Alternatively, one can think of a queueing system where jobs are split into several fragments that are then randomly routed to specific servers in the network to be served in a synchronized fashion. We assume that the system operates on a first-come, first-served basis. The synchronization and service discipline create blocking and idleness among the servers, which leads to a strict stability condition compared with other distributed queueing models. We analyze the stationary waiting time distribution of jobs under a many-server limit and provide exact tail asymptotics. These asymptotics generalize the celebrated Cramér–Lundberg approximation for the single-server queue.

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of $ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} }, u\to \infty , $ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena.

Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.

On the cumulative Parisian ruin of multi-dimensional Brownian motion risk models

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalisations of some of the results derived in Dbicki et al. [(2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and Their Applications 128, 4171–4206]. As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models

Given a set of species whose evolution is represented by a species tree, a gene family is a group of genes having evolved from a single ancestral gene. A gene family evolves along the branches of a species tree through various mechanisms, including—but not limited to—speciation ({mathbb {S}}), gene duplication ({mathbb {D}}), gene loss ({mathbb {L}}), and horizontal gene transfer ({mathbb {T}}). The reconstruction of a gene tree representing the evolution of a gene family constrained by a species tree is an important problem in phylogenomics. However, unlike in the multispecies coalescent evolutionary model that considers only speciation and incomplete lineage sorting events, very little is known about the search space for gene family histories accounting for gene duplication, gene loss and horizontal gene transfer (the {mathbb {D}}{mathbb {L}}{mathbb {T}}-model). In this work, we introduce the notion of evolutionary histories defined as a binary ordered rooted tree describing the evolution of a gene family, constrained by a species tree in the {mathbb {D}}{mathbb {L}}{mathbb {T}}-model. We provide formal grammars describing the set of all evolutionary histories that are compatible with a given species tree, whether it is ranked or unranked. These grammars allow us, using either analytic combinatorics or dynamic programming, to efficiently compute the number of histories of a given size, and also to generate random histories of a given size under the uniform distribution. We apply these tools to obtain exact asymptotics for the number of gene family histories for two species trees, the rooted caterpillar and complete binary tree, as well as estimates of the range of the exponential growth factor of the number of histories for random species trees of size up to 25. Our results show that including horizontal gene transfers induce a dramatic increase of the number of evolutionary histories. We also show that, within ranked species trees, the number of evolutionary histories in the {mathbb {D}}{mathbb {L}}{mathbb {T}}-model is almost independent of the species tree topology. These results establish firm foundations for the development of ensemble methods for the prediction of reconciliations.

Exact L2-Small Ball Asymptotics for Some Durbin Processes

Exact L2-Small Ball Asymptotics for Some Durbin Processes

Extremes of standard multifractional Brownian motion

Let SMBH(t),t∈(0,∞) be a standard multifractional Brownian motion(smBm), where H(t)∈(0,1) is a function of t. In this paper we derive the exact asymptotics of Psupt∈[T1,T2]SMBH(t)>u,u→∞for constants T1,T2≥0 and several forms of H(t).

Informed Trading, Limit Order Book and Implementation Shortfall: Equilibrium and Asymptotics

We propose a static equilibrium model for limit order book where $N\geq 1$ profit-maximizing investors receive an information signal regarding the liquidation value of the asset and execute via a competitive dealer with random initial inventory. While the dealer's initial position plays a role similar to noise traders in Kyle, he trades against a competitive limit order book populated by liquidity suppliers as in Glosten. We show that an equilibrium exists for bounded signal distributions, obtain closed form solutions for Bernoulli-type signals and propose a straightforward iterative algorithm to compute the equilibrium order book for the general case. We obtain the exact analytic asymptotics for the market impact of large trades and show that the functional form depends on the tail distribution of the private signal of the insiders. In particular, the impact follows a power law if the signal has fat tails while the law is logarithmic in case of lighter tails. Moreover, the tail distribution of the trade volume in equilibrium obeys a power law in our model. We find that the liquidity suppliers charge a minimum bid-ask spread that is independent of the amount of `noise' trading but increasing in the degree of informational advantage of insiders in equilibrium. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs. In case of a monopolistic insider we show that the last slice traded against the limit order book is priced at the liquidation value of the asset. However, competition among the insiders leads to aggressive trading causing the aggregate profit to vanish in the limiting case N → ∞. The numerical results also show that the spread increases with the number of insiders keeping the other parameters fixed. Finally, an equilibrium may not exist if the liquidation value is unbounded. We conjecture that existence of equilibrium requires a sufficient amount of competition among insiders if the signal distribution exhibit fat tails.

On the boundary local time measure of super-Brownian motion

In [9] the Hausdorff dimension, $d_{f}$, of $\partial \mathcal {R}$, the topological boundary of the range of super-Brownian motion for dimensions $d=2,3$ was found; $d_{f}=4-2\sqrt {2}$ if $d=2$, and $d_{f}=(9-\sqrt {17})/2$ if $d=3$. We will refine these dimension estimates in a number of ways. If $L^{x}$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal {L}$, called the boundary local time measure, as a rescaling of $L^{x} e^{-\lambda L^{x}} dx$ as $\lambda \to \infty $, thus confirming a conjecture of [19] and further show that the support of $\mathcal {L}$ equals $\partial \mathcal {R}$. This latter result uses a second construction of a boundary local time $\widetilde {\mathcal {L}}$ given in terms of exit measures and we prove that $\widetilde {\mathcal {L}}=c\mathcal {L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal {L}$ in terms of negative dimensional Bessel processes and use them with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial \mathcal {R}$ in [9]. The construction requires a refinement of the $L^{2}$ upper bounds in [19] and [9] to exact $L^{2}$ asymptotics. The methods also refine the left tail bounds for $L^{x}$ in [19] to exact asymptotics. We conjecture that the $d_{f}$-dimensional Minkowski content of $\partial \mathcal {R}$ is equal to the total mass of the boundary local time $\mathcal {L}$ up to some constant.