**A representation of noncommutative BMO spaces**

Publication date: Available online 16 May 2019Source: Statistics & Probability LettersAuthor(s): Ali Talebi, M.S. Moslehian, Ghadir SadeghiAbstractIt is known that the noncommutative Hardy spaces H1(M) and H1max(M) do not coincide, in general. In fact, it may happen that H1(M)⊈H1max(M). It is an interesting question whether the reverse inclusion holds or not. In this note, motivated by this question, we prove that the validity of inequality ‖x‖H1C(M)≤c‖x‖H1max(M) for some c and all martingales x∈H1max(M) gives rise to a certain representation of positive elements of the noncommutative space BMO. Conve...

**Source: **Statistics and Probability Letters - May 17, 2019 **Category: **Statistics **Source Type: **research

**The density flatness phenomenon**

Publication date: Available online 17 May 2019Source: Statistics & Probability LettersAuthor(s): Abbas Alhakim, S. MolchanovAbstractThis paper investigates a curious phenomenon of some positive random variables having a constant density in some initial subinterval of their support. We discuss two methods of obtaining such a random variable with partly flat density. The main method is to sum powers of independent uniform variables with the number of terms matching the power. We show that the limit of this sum is an interesting infinitely divisible distribution and we study its basic properties, find a differential equat...

**Source: **Statistics and Probability Letters - May 17, 2019 **Category: **Statistics **Source Type: **research

**A note on limit theory for mildly stationary autoregression with a heavy-tailed GARCH error process**

Publication date: September 2019Source: Statistics & Probability Letters, Volume 152Author(s): Eunju HwangAbstractA first-order mildly stationary autoregression with a heavy-tailed GARCH error process is considered to study the limit theory for the least squared estimator of the autoregression coefficient ρ=ρn∈[0,1). A Gaussian limit theory is established as ρn converges to the unity as n→∞, with rate condition (1−ρn)n→∞, as in Giraitis and Philips (2006), who have discussed the limit theory in case that errors are martingale difference sequences. This work addresses asy...

**Source: **Statistics and Probability Letters - May 17, 2019 **Category: **Statistics **Source Type: **research

**A note on the coefficients of elliptical random variables**

Publication date: Available online 15 May 2019Source: Statistics & Probability LettersAuthor(s): Tomer ShushiAbstractIt is well-known that the sum of elliptical random variables with constant coefficients is also an elliptical random variable. In this note, we prove that the coefficients can also be random variables, such that the sum of elliptical random variables with random coefficients on a circle is also elliptically distributed. We then examine statistical implications by considering the projection pursuit with the underlying random variables. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 16, 2019 **Category: **Statistics **Source Type: **research

**The naive Bayes classifier for functional data**

Publication date: Available online 14 May 2019Source: Statistics & Probability LettersAuthor(s): Yi-Chen Zhang, Lyudmila SakhanenkoAbstractIn this article, we extended the approach of Dai et al. (2017) to several populations case and compared various other approaches on simulated and real data. The results show that the naive Bayes classifier for functional data is applicable to multi-category classification problems and have preferable finite-sample performance over competitors. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 15, 2019 **Category: **Statistics **Source Type: **research

**Non-asymptotic bounds for percentiles of independent non-identical random variables**

Publication date: Available online 14 May 2019Source: Statistics & Probability LettersAuthor(s): Dong XiaAbstractThis note displays an interesting phenomenon for the percentiles of independent but non-identical random variables. Let X1,…,Xn be independent random variables obeying non-identical continuous distributions and X(1)≥⋯≥X(n) be the corresponding order statistics. For p∈(0,1), we investigate the 100(1−p)%th percentile X(⌊pn⌋) and prove the non-asymptotic bounds for X(⌊pn⌋). In particular, for a wide class of distributions, we discover an intriguing connectio...

**Source: **Statistics and Probability Letters - May 15, 2019 **Category: **Statistics **Source Type: **research

**Reflected backward stochastic partial differential equations with jumps in a convex domain**

Publication date: Available online 13 May 2019Source: Statistics & Probability LettersAuthor(s): Xue YangAbstractThis paper is concerned with a class of multi-dimensional reflected backward stochastic partial differential equations (BSPDE for short), taking values in a convex domain in Rk, which are driven by an infinite dimensional Brownian motion and an independent compensated Poisson random measure. Existence and uniqueness of the solution to this class of reflected BSPDEs are proved. Penalization method plays an important role. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 15, 2019 **Category: **Statistics **Source Type: **research

**Norms of sub-exponential random vectors**

Publication date: Available online 14 May 2019Source: Statistics & Probability LettersAuthor(s): Krzysztof ZajkowskiAbstractWe discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the so-called uniform norm is majorized by norms of coordinates up to some constants. We give an application of other norm to study of chaos in random vectors with sub-exponential coordinates. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 15, 2019 **Category: **Statistics **Source Type: **research

**On a lower bound for the Chung–Diaconis–Graham random process**

Publication date: Available online 9 May 2019Source: Statistics & Probability LettersAuthor(s): Martin HildebrandAbstractConsider the random process on the integers mod p with X0=0 and Xn+1=2Xn+bn(modp) where b0,b1,b2,… are i.i.d. random variables which can have only 1, 0, and −1 as possible values. We show that unless P(bn=0)=1∕2 or P(bn=1)=P(bn=−1)=1∕2, then for some C>1 depending on the probability distribution for bn, at least Clog2p steps are needed to make Xn close to uniformly distributed. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 13, 2019 **Category: **Statistics **Source Type: **research

**Identifying the number of factors using a white noise test**

Publication date: Available online 11 May 2019Source: Statistics & Probability LettersAuthor(s): Shuangbo Li, Li-Xin ZhangAbstractWe propose a new method to estimate the number of factors in a factor model using a new test for vector white noise for high-dimensional cases. We provide the theoretical basis for this method, and the simulation results indicate that the present method outperforms previous methods in some finite sample cases. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 12, 2019 **Category: **Statistics **Source Type: **research

**On a lower bound for the Chung-Diaconis-Graham random process**

Publication date: Available online 9 May 2019Source: Statistics & Probability LettersAuthor(s): Martin HildebrandAbstractConsider the random process on the integers mod p with X0=0 and Xn+1=2Xn+bn(modp) where b0,b1,b2,… are i.i.d. random variables which can have only 1, 0, and −1 as possible values. We show that unless P(bn=0)=1∕2 or P(bn=1)=P(bn=−1)=1∕2, then for some C>1 depending on the probability distribution for bn, at least Clog2p steps are needed to make Xn close to uniformly distributed. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 10, 2019 **Category: **Statistics **Source Type: **research

**Spectral distribution of large generalized random kernel matrices**

Publication date: Available online 9 May 2019Source: Statistics & Probability LettersAuthor(s): Xingyuan ZengAbstractWe consider n×n random kernel matrix An whose entries are of form F(Xi,Xj)−unδij∑kF(Xi,Xk) with F(X,Y)=f(X′Y∕p) or f(|X−Y|2∕p) for i.i.d. random vectors Xi=Σp1∕2Yi∈Rp. Here f is a real-valued function, un is any real number which is allowed to change with n, Σp is a p×p positive semi-definite matrix and the entries of Yi’s are i.i.d. mean 0, variance 1 and have bounded m
(m>4) absolute moments. Explicit limits of spectral distributi...

**Source: **Statistics and Probability Letters - May 10, 2019 **Category: **Statistics **Source Type: **research

**Editorial Board**

Publication date: August 2019Source: Statistics & Probability Letters, Volume 151Author(s): (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 10, 2019 **Category: **Statistics **Source Type: **research

**On general Bayesian inference using loss functions**

Publication date: Available online 8 May 2019Source: Statistics & Probability LettersAuthor(s): Pier Giovanni Bissiri, Stephen G. WalkerAbstractBissiri et al. (2016) propose a framework for general Bayesian inference using loss functions which connec parameters with data, and the updated posterior distribution is characterized through a set of axioms. The result, which is restricted to finite probability spaces, is extended in this paper to spaces which are subsets of the real line. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - May 9, 2019 **Category: **Statistics **Source Type: **research

**Adjusted empirical likelihood method for the tail index of a heavy-tailed distribution**

Publication date: Available online 3 May 2019Source: Statistics & Probability LettersAuthor(s): Yizeng Li, Yongcheng QiAbstractEmpirical likelihood is a well-known nonparametric method in statistics and has been widely applied in statistical inference. The method has been employed by Lu and Peng (2002) to constructing confidence intervals for the tail index of a heavy-tailed distribution. It is demonstrated in Lu and Peng (2002) that the empirical likelihood-based confidence intervals performs better than confidence intervals based on normal approximation in terms of the coverage probability. In general, the empirical ...

**Source: **Statistics and Probability Letters - May 4, 2019 **Category: **Statistics **Source Type: **research

**A conditioned continuous-state branching process with applications**

Publication date: Available online 2 May 2019Source: Statistics & Probability LettersAuthor(s): Rongjuan Fang, Zenghu LiAbstractA supercritical CB-process conditioned on explosion is again a Markov process. We characterize the transition semigroup of the conditioned process by its Laplace transform and by a Doob’s h-transform. The conditioned CB-process is constructed as the strong solution of a stochastic integral equation. We use the distribution of the conditioned process to construct directly the canonical Kuznetsov measure of the CB-process. The later is reconstructed from positive paths picked up by a Poiss...

**Source: **Statistics and Probability Letters - May 4, 2019 **Category: **Statistics **Source Type: **research

**Martingale decomposition and approximations for nonlinearly dependent processes**

Publication date: Available online 30 April 2019Source: Statistics & Probability LettersAuthor(s): Ji Hyung LeeAbstractThis paper proposes a new martingale (MG) decomposition (Gordin, 1969; Hall and Heyde, 1980) for a dependent time series under a predictive dependence measure based on Wu (2005). The decomposition produces a generalized version of the Beveridge–Nelson (BN) lemma (Phillips and Solo, 1992) that accommodates many nonlinear time series, such as GARCH models and threshold autoregressive processes, thereby extending the empirical ambit of the original lemma designed for the linear process. Under this e...

**Source: **Statistics and Probability Letters - May 1, 2019 **Category: **Statistics **Source Type: **research

**Exact strong laws of large numbers for ratios of the smallest order statistics**

Publication date: Available online 29 April 2019Source: Statistics & Probability LettersAuthor(s): Przemysław Matuła, Paweł Kurasiński, André AdlerAbstractLet Xn,kn∈N,1≤k≤kn be an array of independent and identically distributed random variables. We take the smallest order statistics Xn,(1) and Xn,(2) in each row and consider the ratios Rn=Xn,(2)∕Xn,(1). Then we study the almost sure convergence of weighted sums of these ratios in two cases: kn→∞ and when kn=K is fixed. Our theorems are proved under very mild conditions and they extend the results of all the papers on this topic that ...

**Source: **Statistics and Probability Letters - April 29, 2019 **Category: **Statistics **Source Type: **research

**Characterization of distributions of Q-independent random variables on locally compact Abelian groups**

Publication date: Available online 27 April 2019Source: Statistics & Probability LettersAuthor(s): Margaryta MyronyukAbstractLet X be a second countable locally compact Abelian group. We prove some group analogues of the Skitovich–Darmois, Heyde and Kac–Bernstein characterisation theorems for Q-independent random variables taking values in the group X. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 28, 2019 **Category: **Statistics **Source Type: **research

**Abundance estimation based on optimal estimating function with missing covariates in capture–recapture studies**

Publication date: Available online 23 April 2019Source: Statistics & Probability LettersAuthor(s): Yang Liu, Xiuzhen Zhang, Mengke Li, Guanfu Liu, Lin ZhuAbstractFor the abundance parameter in capture–recapture studies with missing covariates, we propose an optimal-estimating-function-based estimator and develop a bootstrap procedure to construct the confidence interval. We illustrate our methods through simulations and the yellow-bellied prinia data. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 27, 2019 **Category: **Statistics **Source Type: **research

**Nonexistence of an unbiased estimating function for the Cox model**

Publication date: Available online 25 April 2019Source: Statistics & Probability LettersAuthor(s): Xuan Yao, Eric SludAbstractIt is proved under broad regularity conditions that there is no unbiased estimating function depending only on the structural parameter (the coefficients) in the Cox proportional hazards model with nuisance survival hazard parameters and possibly right-censored survival times. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 27, 2019 **Category: **Statistics **Source Type: **research

**On sharp nonparametric estimation of differentiable functions**

Publication date: Available online 25 April 2019Source: Statistics & Probability LettersAuthor(s): Sam EfromovichAbstractSharp minimax nonparametric estimation is well known for functions from a Sobolev ellipsoid which is defined via Fourier coefficients of differentiable functions with specific boundary conditions. The theory is based on a renown lower bound of Pinsker (1980) and an adaptive estimator that attains it. This paper solves a long-standing problem of adaptive estimation without assuming boundary conditions. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 26, 2019 **Category: **Statistics **Source Type: **research

**Abundance estimation based on optimal estimating function with missing covariates in capture-recapture studies**

Publication date: Available online 23 April 2019Source: Statistics & Probability LettersAuthor(s): Yang Liu, Xiuzhen Zhang, Mengke Li, Guanfu Liu, Lin ZhuAbstractFor the abundance parameter in capture-recapture studies with missing covariates, we propose an optimal-estimating-function-based estimator and develop a bootstrap procedure to construct the confidence interval. We illustrate our methods through simulations and the yellow-bellied prinia data. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 25, 2019 **Category: **Statistics **Source Type: **research

**Nonparametric relative regression under random censorship model**

Publication date: Available online 18 April 2019Source: Statistics & Probability LettersAuthor(s): Salah Khardani, Yousri SlaouiAbstractIn this paper we define and study a new estimator of the regression function when the response random variable is subject to random right-censoring. The estimator is constructed by minimizing the mean squared relative error of the regression operator where outlier data are present and the response variable of the model is positive. Under classical conditions we establish the uniform consistency with rate over a compact set and asymptotic normality of the estimator suitably normalized. ...

**Source: **Statistics and Probability Letters - April 19, 2019 **Category: **Statistics **Source Type: **research

**On a flexible construction of a negative binomial model**

Publication date: Available online 17 April 2019Source: Statistics & Probability LettersAuthor(s): Fabrizio Leisen, Ramsés H. Mena, Freddy Palma Mancilla, Luca RossiniAbstractThis work presents a construction of stationary Markov models with negative-binomial marginal distributions. A simple closed form expression for the corresponding transition probabilities is given, linking the proposal to well-known classes of birth and death processes and thus revealing interesting characterizations. The advantage of having such closed form expressions is tested on simulated and real data. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 18, 2019 **Category: **Statistics **Source Type: **research

**Sharpness for inhomogeneous percolation on quasi-transitive graphs**

Publication date: Available online 16 April 2019Source: Statistics & Probability LettersAuthor(s): Thomas Beekenkamp, Tim HulshofAbstractIn this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different “types” of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponen...

**Source: **Statistics and Probability Letters - April 17, 2019 **Category: **Statistics **Source Type: **research

**Applications of the fundamental matrix to mean absorption and conditional mean absorption problems**

Publication date: Available online 12 April 2019Source: Statistics & Probability LettersAuthor(s): Evan MillikenAbstractA proof of a general theorem for the calculation of conditional mean duration of a finite absorbing discrete time Markov chain is presented. In the simplest case, this result is equivalent to one suggested in the book of Kemeny and Snell (1976). In addition, we prove that the mean duration and mean conditional duration of a finite absorbing continuous time Markov chain can be calculated via the fundamental matrix of the embedded discrete time chain. These results are also extended to certain non-absor...

**Source: **Statistics and Probability Letters - April 13, 2019 **Category: **Statistics **Source Type: **research

**The distribution of the minimum of a positive sample**

Publication date: Available online 8 April 2019Source: Statistics & Probability LettersAuthor(s): Christopher S. Withers, Saralees NadarajahAbstractWe give expansions for the distribution function, density function and moments of the sample minimum when sampling from a distribution on 0,x2 that is nearly analytic at zero. Similar results are given for the sample maximum when sampling from a distribution on x1,x2 that is nearly analytic at x2. When these distributions are analytic, the expansions are in inverse powers of the sample size n. If not, they require a double expansion. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 10, 2019 **Category: **Statistics **Source Type: **research

**Editorial Board**

Publication date: July 2019Source: Statistics & Probability Letters, Volume 150Author(s): (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 7, 2019 **Category: **Statistics **Source Type: **research

**An elementary proof of de Finetti’s theorem**

We present an new, elementary proof of de Finetti’s Theorem. The purpose of this paper is to make this theorem accessible to a broader community through an essentially self-contained proof. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 4, 2019 **Category: **Statistics **Source Type: **research

**Exact convergence rate in the local central limit theorem for a lattice branching random walk on Zd**

Publication date: Available online 3 April 2019Source: Statistics & Probability LettersAuthor(s): Zhiqiang GaoAbstractConsider a branching random walk, where the branching mechanism is governed by a Galton–Watson process and the migration is governed by a lattice random walk on Zd. Under the mild moment conditions for the underlying branching mechanism and migration laws, we figure out the exact convergence rate of the local limit theorem for the counting measure which counts the number of particles of generation n in a given set. Our result extends the previous one obtained by Chen (2001) for a simple branching ...

**Source: **Statistics and Probability Letters - April 4, 2019 **Category: **Statistics **Source Type: **research

**The asymmetric log-Laplace distribution as a limiting case of the generalized beta distribution**

Publication date: Available online 1 April 2019Source: Statistics & Probability LettersAuthor(s): Joshua D. Higbee, Jonathan E. Jensen, James B. McDonaldAbstractThe asymmetric log-Laplace (ALL) and the generalized beta distribution of the second kind (GB2) have been used in many applications. We demonstrate that the ALL is a limiting case of the GB2 and examine their ability to model stock returns. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 2, 2019 **Category: **Statistics **Source Type: **research

**Jensen inequality for superlinear expectations**

Publication date: Available online 2 April 2019Source: Statistics & Probability LettersAuthor(s): Qian LinAbstractIn this note, we investigate Jensen inequality for superlinear expectations and its application in economics. Superlinear expectations are defined via G-expectations. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - April 2, 2019 **Category: **Statistics **Source Type: **research

**Consistent estimation of residual variance with random forest Out-Of-Bag errors**

Publication date: Available online 1 April 2019Source: Statistics & Probability LettersAuthor(s): Burim Ramosaj, Markus PaulyAbstractThe issue of estimating residual variance in regression models with unknown and eventually complex link-function is still an open problem. Predictions of such outcomes are usually conducted by black-box procedures without seriously restricting the link-function class. However, quantifying uncertainty by means of residual variance estimators is of primary interest in many practical applications, e.g. as a primary step towards the construction of prediction intervals. Here, we consider this...

**Source: **Statistics and Probability Letters - April 2, 2019 **Category: **Statistics **Source Type: **research

**p-conformal maps on the triangular lattice**

Publication date: Available online 28 March 2019Source: Statistics & Probability LettersAuthor(s): Jiro Akahori, Yuuki Ida, Greg MarkowskyAbstractIn Akahori and Ida (2014), p-conformal (or Parisian-conformal) maps on the triangular lattice were defined. The definition of p-conformality is nonstandard in comparison to ordinary discrete derivatives, but was seen to be natural in connection with a particular type of random walk, the Parisian random walk. In this note, we establish the fact that the only p-conformal polynomials in z and z̄ on the triangle lattice are linear combinations of 1, z and z2−z̄, but that ...

**Source: **Statistics and Probability Letters - March 29, 2019 **Category: **Statistics **Source Type: **research

**Compatible matrices of Spearman’s rank correlation**

Publication date: Available online 29 March 2019Source: Statistics & Probability LettersAuthor(s): Bin Wang, Ruodu Wang, Yuming WangAbstractIn this paper, we provide a negative answer to a long-standing open problem on the compatibility of Spearman’s rho matrices. Following an equivalence of Spearman’s rho matrices and linear correlation matrices for dimensions up to 9 in the literature, we show non-equivalence for dimensions 12 or higher. In particular, we connect this problem with the existence of a random vector under some linear projection restrictions in two characterization results. (Source: Statistic...

**Source: **Statistics and Probability Letters - March 29, 2019 **Category: **Statistics **Source Type: **research

**Two part envelopes for rejection sampling of some completely random measures**

Publication date: Available online 27 March 2019Source: Statistics & Probability LettersAuthor(s): Jim GriffinAbstractThis paper shows that rejection sampling with a two-piece Lévy intensity envelope can outperform both the Ferguson-Klass algorithm and previously proposed envelopes for simulating realisations of completely random measures typically used in Bayesian nonparametric statistics. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 28, 2019 **Category: **Statistics **Source Type: **research

**Large and moderate deviation principles for recursive kernel estimators of a regression function for spatial data defined by stochastic approximation method**

Publication date: Available online 22 March 2019Source: Statistics & Probability LettersAuthor(s): Salim Bouzebda, Yousri SlaouiAbstractIn the present paper, we are mainly concerned with a family of kernel type estimators based upon spatial data. More precisely, we establish large and moderate deviations principles for the recursive kernel estimators of a regression function for spatial data defined by the stochastic approximation algorithm. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 24, 2019 **Category: **Statistics **Source Type: **research

**Nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises**

Publication date: Available online 22 March 2019Source: Statistics & Probability LettersAuthor(s): Xuekang Zhang, Haoran Yi, Huisheng ShuAbstractThe present paper deals with the problem of nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises based on continuous-time observation. Under some certain conditions, we derive the uniform consistency and the rate of convergence of the nonparametric estimator. Besides, the asymptotic distribution of the estimator in our setting is shown to be a stable distribution. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 24, 2019 **Category: **Statistics **Source Type: **research

**Covariance functions on spheres cross time: Beyond spatial isotropy and temporal stationarity**

Publication date: Available online 21 March 2019Source: Statistics & Probability LettersAuthor(s): Anne Estrade, Alessandra Fariñas, Emilio PorcuAbstractSpectral representations uniquely define the covariance functions associated to random fields defined over spheres or spheres cross time. Covariance functions on spheres cross time are usually modelled under the assumptions of either spatial isotropy or axial symmetry, and the assumption of temporal stationarity. This paper goes beyond these assumptions. In particular, we consider the problem of spatially anisotropic covariance functions on spheres. The crux of ...

**Source: **Statistics and Probability Letters - March 23, 2019 **Category: **Statistics **Source Type: **research

**Large rank-based models with common noise**

Publication date: Available online 22 March 2019Source: Statistics & Probability LettersAuthor(s): Praveen Kolli, Andrey SarantsevAbstractFor large systems of Brownian particles interacting through their ranks introduced in (Banner et al., 2005), the empirical cumulative distribution function satisfies a porous medium PDE. However, when we introduce a common noise, the limit is no longer deterministic. Instead, we show that this limit is a solution of a stochastic PDE related to this porous medium PDE. This stochastic PDE is somewhat similar to the equations developed for conservation laws with rough stochastic fl...

**Source: **Statistics and Probability Letters - March 23, 2019 **Category: **Statistics **Source Type: **research

**Intrinsic metric and one-dimensional diffusions**

Publication date: Available online 19 March 2019Source: Statistics & Probability LettersAuthor(s): Wenjie Sun, Jiangang YingAbstractOne-dimensional strongly local and regular Dirichlet forms which are irreducible can always be characterized by the so-called scale function s and speed measure m. In this paper we derive the intrinsic metric of such a Dirichlet form in terms of s and m. As an application, we give a new characterization of regular Dirichlet subspaces and extensions of Brownian motion, comparing to Fang et al. (2005) and Li and Ying (2019). Finally two examples on volume doubling property of regular Di...

**Source: **Statistics and Probability Letters - March 20, 2019 **Category: **Statistics **Source Type: **research

**On the first positive and negative excursion exceeding a given length**

Publication date: Available online 19 March 2019Source: Statistics & Probability LettersAuthor(s): Roberta Sirovich, Luisa TestaAbstractFor a one-dimensional diffusion process X, we derive the Laplace transform and the moments of the first time at which the age of an excursion above (or below) the level x is longer than u. The result is then illustrated for diffusion processes that are found relevant in applications. In the context of pricing Parisian options, the Brownian motion and the Geometric Brownian motion are considered and the Laplace transform can be made explicit and explicit expression for the moments can b...

**Source: **Statistics and Probability Letters - March 20, 2019 **Category: **Statistics **Source Type: **research

**Observing a Lévy process up to a stopping time**

Publication date: Available online 18 March 2019Source: Statistics & Probability LettersAuthor(s): Matija VidmarAbstractIt is proved that the law of a possibly killed Lévy process X, seen up to and including (resp. up to strictly before) a stopping time, determines already the law of X (resp. up to a compound Poisson component and killing). (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 19, 2019 **Category: **Statistics **Source Type: **research

**Best linear predictor of a C[0,1]-valued functional autoregressive process**

Publication date: Available online 14 March 2019Source: Statistics & Probability LettersAuthor(s): Meryem Kada Kloucha, Tahar MouridAbstractWe establish exponential bounds and the a.s. convergence of the best linear predictor BLP of a C[0,1]-valued autoregressive process. Simulation studies on real series illustrate the performance of the BLP and show competitive results. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 15, 2019 **Category: **Statistics **Source Type: **research

**Limit theory for moderate deviation from Integrated GARCH processes**

Publication date: Available online 14 March 2019Source: Statistics & Probability LettersAuthor(s): Yubo TaoAbstractThis paper develops the limit theory of the GARCH(1,1) process that moderately deviates from IGARCH process towards both stationary and explosive regimes. The asymptotic theory extends Berkes et al. (2005) by allowing the parameters to have a slower rate of convergence. The results can be applied to unit root test for processes with mildly-integrated GARCH innovations (e.g. Boswijk (2001), Cavaliere and Taylor (2007, 2009)) and deriving limit theory of estimators for models involving mildly-integrated...

**Source: **Statistics and Probability Letters - March 15, 2019 **Category: **Statistics **Source Type: **research

**Squared-norm empirical processes**

Publication date: Available online 10 March 2019Source: Statistics & Probability LettersAuthor(s): Vincent Q. Vu, Jing LeiAbstractThis note extends a result of Mendelson on the supremum of a quadratic process to squared norms of functions taking values in a Banach space. Our method of proof is a reduction by a symmetrization argument and simple observation about the additivity of the generic chaining functional. We demonstrate an application to positive linear functionals of the sample covariance matrix and the apparent variance explained by principal components analysis (PCA). (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 11, 2019 **Category: **Statistics **Source Type: **research

**On strong stationary times and approximation of Markov chain hitting times by geometric sums**

Publication date: Available online 11 March 2019Source: Statistics & Probability LettersAuthor(s): Fraser DalyAbstractFor discrete-time, ergodic Markov chains started from stationarity, Fill and Lyzinski (2014) showed that hitting times may sometimes be represented as geometric sums. We extend this, giving explicit bounds in the approximation of hitting times by appropriately chosen geometric sums. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 11, 2019 **Category: **Statistics **Source Type: **research

**The Csörgő–Révész moduli of non-differentiability of fractional Brownian motion**

Publication date: Available online 5 March 2019Source: Statistics & Probability LettersAuthor(s): Wensheng Wang, Yimin XiaoAbstractWe establish the exact moduli of non-differentiability of fractional Brownian motion. As an application of the result, we prove that the uniform Hölder condition for the maximum local times of fractional Brownian motion obtained in Xiao (1997) is optimal. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 10, 2019 **Category: **Statistics **Source Type: **research

**Change-point analysis using logarithmic quantile estimation**

We present a new approach for estimating quantiles for change-point problems, specifically for Pettitt’s rank test (1979). We use the logarithmic quantile estimation procedure introduced by Thangavelu (2005), which is based on the concept of the almost sure limit theorem. Numerical results for small data sets and simulated data are given. (Source: Statistics and Probability Letters)

**Source: **Statistics and Probability Letters - March 8, 2019 **Category: **Statistics **Source Type: **research