Functional Limit Theorem Research Articles

A functional limit theorem for limit order books with state dependent price dynamics

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

Ergodicity of inhomogeneous Markov chains through asymptotic pseudotrajectories

In this work, we consider an inhomogeneous (discrete time) Markov chain and are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a homogeneous (continuous time) Markov process. Renowned examples such as a bandit algorithms, weighted random walks or decreasing step Euler schemes are included in our framework. Our results are related to functional limit theorems, but the approach differs from the standard "Tightness/Identification" argument; our method is unified and based on the notion of pseudotrajectories on the space of probability measures.

P(𝜙)1-process for the spin-boson model and a functional central limit theorem for associated additive functionals

We construct a random process with stationary increments associated to the Hamiltonian of the spin-boson model consisting of a component describing the spin and a component given by a Schwartz distribution-valued Ornstein-Uhlenbeck process describing the boson field. We use a path integral representation of the Hamiltonian to prove a functional central limit theorem for additive functionals, and derive explicit expressions of the diffusion constant for specific functionals.

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

A functional limit theorem for irregular SDEs

Soit $X_{1},X_{2},\ldots$ une suite de variables aleatoires independantes avec esperance $E(X_{i})=0$, et $Y^{N}_{k+1}=Y^{N}_{k}+a_{N}(Y^{N}_{k})X_{k+1}$ une marche aleatoire renormalisee avec une fonction $a_{N}:\mathbb{R}\to\mathbb{R}_{+}$. On montre, sous certaines conditions legeres sur la loi de $X_{i}$, que l’on peut choisir le facteur $a_{N}$ d’une facon que $(Y^{N}_{\lfloor Nt\rfloor})_{t\in\mathbb{R}_{+}}$ converge en loi, quand $N$ tend vers l’infini, vers une diffusion $(M_{t})_{t\in\mathbb{R}_{+}}$ etant la solution d’une equation differentielle stochastique avec des coefficients irreguliers. A cet effet, nous plongeons la marche aleatoire renormalisee dans la diffusion $M$ par une suite de temps d’arret ayant un pas de temps avec esperance $1/N$.

Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].

Functional limit theorems for a new class of non-stationary shot noise processes

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.

Path to survival for the critical branching processes in a random environment

Abstract A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

On the functional and local limit theorems for Markov modulated compound Poisson processes

We study a class of Markov-modulated compound Poisson processes whose arrival rates and the compound random variables are both modulated by a stationary finite-state Markov process. The compound random variables are i.i.d. in each state of the Markov process, while having a distribution depending on the state of the Markov process. We prove a functional central limit theorem and local limit theorems under appropriate scalings of the arrival process, compound random variables and underlying Markov process.

Central limit theorem for functionals of a generalized self-similar Gaussian process

We consider a class of self-similar, continuous Gaussian processes that do not necessarily have stationary increments. We prove a version of the Breuer–Major theorem for this class, that is, subject to conditions on the covariance function, a generic functional of the process increments converges in law to a Gaussian random variable. The proof is based on the Fourth Moment Theorem. We give examples of five non-stationary processes that satisfy these conditions.

Functional limit theorems for Galton–Watson processes with very active immigration

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979).

A conditional functional limit theorem for decomposable branching processes with two types of particles

Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time Nt, 0 ≤ t < ∞, given that the number of particles of the first type appearing in the process during its evolution is N.

Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and Leon (Extremes 3(1):57–86, 2000), Kratz and Leon (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and Leon in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincare characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.

A functional limit theorem for random processes with immigration in the case of heavy tails

Let $(X_k,\xi_k)_{k\in \mathbb {N}}$ be a sequence of independent copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb {R}}$ is defined as the a.s. finite sum $Y(u)=\sum_{k\geq0}X_{k+1}(u- \xi_1-\cdots-\xi_k)1\mkern-4.5mu\mathrm{l}_{\{\xi_1+\cdots+\xi_k\leq u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\geq 0}$, as $t\to\infty$, when the law of $\xi$ belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(0,1)$, and the process $X$ oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\geq0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty)$ to a fractionally integrated inverse stable subordinator.

A note on the limit theory of a Dickey–Fuller unit root test with heavy tailed innovations

We are occupied with the limit theory of the OLSE and of a subsequent Dickey–Fuller test when the unit root process has heavy tailed and dependent innovations that do not possess moments of order α for some α∈0,2. The innovation process has the form of a “martingale-type” transform constructed as a pointwise product between an iid sequence in the domain of attraction of an α stable distribution with a non existing α moment, for some α∈0,2, and a positive scaling mixing sequence that has a slowly varying at infinity truncated α moment. We derive a functional limit theorem with complex rates and limits that depend on Levy α-stable processes. The OLSE remains superconsistent with rate n, and the limiting distribution is a functional of the previous process. When α=2 we recover the standard Dickey–Fuller distribution.

Small drift limit theorems for random walks

AbstractWe show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a&gt;0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

Functional limit theorems for the maxima of perturbed random walk and divergent perpetuities in the M 1-topology

Let (ξ1, η1), (ξ2, η2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J1-topology on the Skorokhod space of \(n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})\) was proved under the assumption that contributions of \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) to the limit are comparable and that n−1/2(ξ1+… + ξ[n⋅]) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ1+… + ξ[n⋅], properly normalized without centering, is attracted to a centered stable Levy process, a process with jumps. As a consequence, weak convergence normally holds in the M1-topology. We also provide sufficient conditions for the J1-convergence. For completeness, less interesting situations are discussed when one of the sequences \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) dominates the other. An application of our main results to divergent perpetuities with positive entries is given.

On transient and stationary regimes for multi-channel networks with periodic inputs

In this paper, a multi-channel queuing network with a periodic input flow is considered. The multivariate service process is studied in transient and stationary regimes as the number of calls at network nodes. A special form of Jackson’s law is obtained. In heavy traffic regime, functional limit theorems of diffusion approximation type are formulated under the condition of convergence in the uniform topology. A limit non-Markov Gaussian process is constructed and its correlation characteristics are represented explicitly via the network parameters.

On functional limit theorems for the cumulative times in alternating renewal processes

We provide new proofs for two functional central limit theorems, and prove strong approximations for the cumulative “on” times in alternating renewal processes. The proofs rely on a first-passage-time representation of the cumulative “on” time process. As an application, we establish strong approximations for the queueing process in a single-server fluid queue with “on–off” sources.

Limit theorems for functionals of Gaussian vectors

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.