Probab Stat Research Articles

Extremal index: estimation and resampling

The duration of extremes in time leads to a phenomenon known as clustering of high values, with a strong impact on risk assessment. The extremal index is a measure developed within Extreme Value Theory that quantifies the degree of clustering of high values. In this work we will consider the cycles estimator introduced in Ferreira and Ferreira (Ann Inst Henri Poincare Probab Stat 54(2):587–605, 2018). A reduced bias estimator based on the Jackknife methodology will be presented. The bootstrap technique will also be considered in the inference and will allow to obtain confidence intervals. The performance will be analyzed based on simulation. We found our proposal effective in reducing bias and it compares favorably with some well-known methods. An application of the methods to real data will also be presented.

Improving the bias of a pseudo-maximum likelihood estimate of the extreme value index by k-records

The paper focusses on the estimation of the extreme value index in terms of k-records based on a maximum likelihood approach, which is suggested recently by Louzaoui and El Arrouchi (J Probab Stat, 2020). Its asymptotic normality is well investigated in order to propose a bias correction while ensuring that the new estimator becomes asymptotically unbiased and still normal. Some numerical studies are also provided in order to show how the proposed estimators behave in practice.

A Large Deviation Principle for the Stochastic Heat Equation with General Rough Noise

We study the Freidlin–Wentzell large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise: $$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$where \(\dot{W}\) is white in time and fractional in space with Hurst parameter \(H\in \left( \frac{1}{4},\frac{1}{2}\right) \). Recently, Hu and Wang (Ann Inst Henri Poincaré Probab Stat 58(1):379–423, 2022) have studied the well-posedness of this equation without the technical condition of \(\sigma (0)=0\) which was previously assumed in Hu et al. (Ann Probab 45(6):4561–4616, 2017). We adopt a new sufficient condition proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) for the weak convergence criterion of the large deviation principle.

Large deviation principles for a 2D liquid crystal model with jump noise

We consider a stochastic 2D liquid crystal model with a multiplicative noise of Lévy type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces of jump type. We derive a large deviation principle for the model. The proof is based on the weak convergence method introduced in [Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincar Probab Stat. 2011;47(3):725–747].

Discretely shrinking targets in moduli space

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic \(\mathrm {SL}(2,\mathbb {R})\)–invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer (Geom Funct Anal 27:1257–1287, 2017) and generalize theorems of Aimino, Nicol, and Todd (Ann Inst Henri Poincaré Probab Stat 53:1371–1401, 2017).

Interacting particle systems and Jacobi style identities

We consider the family of nearest neighbour interacting particle systems on mathbb {Z} allowing 0, 1 or 2 particles at a site. We parametrise a wide subfamily of processes exhibiting product blocking measure and show how this family can be “stood up” in the sense of Balázs and Bowen (Ann Inst H Poincaré Probab Stat 54(1):514–528, 2018). By comparing measures, we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a 2-repetition condition. By specialising to specific processes, we produce two variable identities that are shown to relate to Jacobi triple product and various other identities of combinatorial significance. The family of k-exclusion processes for arbitrary k are also considered and are shown to give similar Jacobi style identities relating to counting generalised Frobenius partitions with a k-repetition condition.

Lower Deviation Probabilities for Level Sets of the Branching Random Walk

Given a supercritical branching random walk $$\{Z_n\}_{n\ge 0}$$ on $${\mathbb {R}}$$ , let $$Z_n(A)$$ be the number of particles located in a set $$A\subset {\mathbb {R}}$$ at generation n. It is known from Biggins (J Appl Probab 14:630–636, 1977) that under some mild conditions, for $$\theta \in [0,1)$$ , $$n^{-1}\log Z_n([\theta x^* n,\infty ))$$ converges almost surely to $$\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*)$$ as $$n\rightarrow \infty $$ , where $$x^*$$ is the speed of the maximal position of $$\{Z_n\}_{n\ge 0}$$ and $$I(\cdot )$$ is the large deviation rate function of the underlying random walk. In this work, we investigate its lower deviation probabilities, in other words, the convergence rates of $$\begin{aligned} {\mathbb {P}}\left( Z_n([\theta x^* n,\infty ))<e^{an}\right) \end{aligned}$$ as $$n\rightarrow \infty $$ , where $$a\in [0,\log \left( {\mathbb {E}}[Z_1({\mathbb {R}})]\right) -I(\theta x^*))$$ . Our results complete those in Chen and He (Ann Institut Henri Poincare Probab Stat 56:2507–2539, 2020), Gantert and Höfelsauer (Electron Commun Probab 23(34):1–12, 2018) and Öz (Latin Am J Probab Math Stat 17:711–731, 2020).

Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $$L^p(\Omega )$$ -solution exists for all $$p\ge 2$$ . In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al. (Inst Henri Poincaré Probab Stat. To appear, 2021). In the case when there exists only a local solution, we determine the precise deterministic time, $$T_2$$ , before which a unique $$L^2(\Omega )$$ -solution exists, but after which the series corresponding to the $$L^2(\Omega )$$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.

Non-Hermitian Random Matrices with a Variance Profile (II): Properties and Examples

For each n, let \(A_n=(\sigma _{ij})\) be an \(n\times n\) deterministic matrix and let \(X_n=(X_{ij})\) be an \(n\times n\) random matrix with i.i.d. centered entries of unit variance. In the companion article (Cook et al. in Electron J Probab 23:Paper No. 110, 61, 2018), we considered the empirical spectral distribution \(\mu _n^Y\) of the rescaled entry-wise product $$\begin{aligned} Y_n = \frac{1}{\sqrt{n}} A_n\odot X_n = \left( \frac{1}{\sqrt{n}} \sigma _{ij}X_{ij}\right) \end{aligned}$$and provided a deterministic sequence of probability measures \(\mu _n\) such that the difference \(\mu ^Y_n - \mu _n\) converges weakly in probability to the zero measure. A key feature in Cook et al. (2018) was to allow some of the entries \(\sigma _{ij}\) to vanish, provided that the standard deviation profiles \(A_n\) satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence \((\mu _n)\), described by a family of Master Equations. We consider these equations in important special cases such as sampled variance profiles \(\sigma ^2_{ij} = \sigma ^2\left( \frac{i}{n}, \frac{j}{n} \right) \) where \((x,y)\mapsto \sigma ^2(x,y)\) is a given function on \([0,1]^2\). Associated examples are provided where \(\mu _n^Y\) converges to a genuine limit. We study \(\mu _n\)’s behavior at zero. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al. (Ann Appl Probab 28(1):148–203, 2018; Ann Inst Henri Poincaré Probab Stat 55(2):661–696, 2019), we prove that, except possibly at the origin, \(\mu _n\) admits a positive density on the centered disc of radius \(\sqrt{\rho (V_n)}\), where \(V_n=(\frac{1}{n} \sigma _{ij}^2)\) and \(\rho (V_n)\) is its spectral radius.

On Stein’s Factors for Poisson Approximation in Wasserstein Distance with Nonlinear Transportation Costs

We establish various bounds on the solutions to a Stein equation for Poisson approximation in the Wasserstein distance with nonlinear transportation costs. The proofs are a refinement of those in Barbour and Xia (Bernoulli 12:943–954, 2006) using the results in Liu and Ma (Ann Inst H Poincaré Probab Stat 45:58–69, 2009). As a corollary, we obtain an estimate of Poisson approximation error measured in the \(L^2\)-Wasserstein distance.

The Gorin–Shkolnikov Identity and Its Random Tree Generalization

In a recent pair of papers, Gorin and Shkolnikov (Ann Probab 46: 2287–2344, 2018) and Hariya (Electron Commun Probab 21: 6, 2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance $$\frac{1}{12}$$ 1 12 . Lamarre and Shkolnikov generalized this to Brownian bridges (Lamarre and Shkolnikov in Ann Inst Henri Poincaré Probab Stat 55: 1402–1438, 2019) and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on n vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous results for a variety of other related models using stochastic calculus.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

AbstractWe establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.

A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; gamma =sqrt{8/3}); and planar maps weighted by the number of different spanning trees (gamma =sqrt{2}), bipolar orientations (gamma =sqrt{4/3}), or Schnyder woods (gamma =1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of gamma -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when gamma =sqrt{8/3}, we instead deduce estimates for the sqrt{8/3}-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

Improved Regression-Cum-Ratio Estimators Using Information on Two Auxiliary Variables Dealing with Subsampling Technique of Non-Response

In the present study, we have suggested some improved regression-cum-ratio-type estimators to estimate the population mean using two auxiliary variables in two different situations of non-response. The bias and mean square error are obtained under large sample approximation. After analysis of a numerical illustration, it has been found that the proposed class of estimators is more efficient than Hansen and Hurwitz (J Am Stat Assoc 41:517–529, 1946) usual unbiased estimator, conventional ratio and regression estimators, Singh and Kumar (Braz J Probab Stat 25(2):205–217, 2011) estimators, Muneer et. al. (Commun Stat Theory Methods 46(5):2181–2192, 2017) estimators, Kumar et. al. (J Stat Comput Simul 88(18):3694–3707, 2018) estimators and Akingbade and Okafer (Pak J Stat Oper Res 15(2):329–340, 2019) estimators. We also consider a simulation study under which the estimated performance of the proposed class of estimators is evaluated.

Dimension transformation formula for conformal maps into the complement of an SLE curve

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of {mathbb {R}} and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an hbox {SLE}_kappa curve for kappa not =4. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes and Vargas (ESAIM Probab Stat 15:358–371, 2011) and the KPZ formula of Gwynne et al. (Ann Probab, 2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an hbox {SLE}_kappa curve for kappa in (0,4)cup (4,8) and the dimension of the same set with respect to the gamma -quantum natural parameterization of the curve induced by an independent Gaussian free field, gamma = sqrt{kappa }wedge (4/sqrt{kappa }).

Markov Semi-groups Associated with the Complex Unimodular Group $$Sl(2,{\mathbb {C}})$$ S l ( 2 , C )

In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group $$SL(2,{\mathbb {C}})$$ to two commutative subalgebras of its universal $$C^{\star }$$ -algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $$-\,1$$ , and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $$Mp(4,{\mathbb {R}})$$ and to the Landau operator in the complex plane.

Recursive Kernel Density Estimation and Optimal Bandwidth Selection Under α: Mixing Data

In this paper, we extend the work of Slaoui (J Probab Stat, https://doi.org/10.1155/2014/739640 , 2014) to the case of strong mixing data. Then, we study the properties of these estimators and compare them with Rosemblatt’s nonrecursive estimator. The bias, variance and MISE are computed explicitly. Using a selected bandwidth and a special stepsize, we showed that the proposed recursive estimators allowed us to obtain quite better results compared to the nonrecursive density estimator under α-mixing condition in terms of estimation error and much better in terms of computational costs.

Approach to the Steady State in Kinetic Models with Thermal Reservoirs at Different Temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in Carlen et al. (Braz J Probab Stat 29:372–386, 2015), we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilistic techniques, and is quite different form the analysis in Carlen et al. (Braz J Probab Stat 29:372–386, 2015) that was based on contraction mapping methods.

Near Critical Preferential Attachment Networks have Small Giant Components

Preferential attachment networks with power law exponent tau >3 are known to exhibit a phase transition. There is a value rho _{mathrm{c}}>0 such that, for small edge densities rho le rho _{mathrm{c}} every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities rho >rho _{mathrm{c}} there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp (-c/ sqrt{rho -rho _{mathrm{c}}}) for an explicit constant c>0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).

The cut‐tree of large trees with small heights

AbstractWe destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017