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The Malliavin-Stein method for Hawkes functionals Hawkes泛函的Malliavin-Stein方法
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-04-04 DOI: 10.30757/alea.v19-52
C. Hillairet, Lorick Huang, Mahmoud Khabou, Anthony Reveillac
{"title":"The Malliavin-Stein method for Hawkes functionals","authors":"C. Hillairet, Lorick Huang, Mahmoud Khabou, Anthony Reveillac","doi":"10.30757/alea.v19-52","DOIUrl":"https://doi.org/10.30757/alea.v19-52","url":null,"abstract":". In this paper, following Nourdin-Peccati’s methodology, we combine the Malliavin calculus and Stein’s method to provide general bounds on the Wasserstein distance between the law of functionals of a compound Hawkes process and the one of a Gaussian random variable. To achieve this, we rely on the Poisson imbedding representation of a Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing a quantitative Central Limit Theorem","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Spread of Infection over P.A. random graphs with edge insertion 带边插入的P.A.随机图上的感染传播
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-03-30 DOI: 10.30757/alea.v19-50
C. Alves, Rodrigo Ribeiro
{"title":"Spread of Infection over P.A. random graphs with edge insertion","authors":"C. Alves, Rodrigo Ribeiro","doi":"10.30757/alea.v19-50","DOIUrl":"https://doi.org/10.30757/alea.v19-50","url":null,"abstract":"In this work we investigate a bootstrap percolation process on random graphs generated by a random graph model which combines preferential attachment and edge insertion between previously existing vertices. The probabilities of adding either a new vertex or a new connection between previously added vertices are time dependent and given by a function $f$ called the edge-step function. We show that under integrability conditions over the edge-step function the graphs are highly susceptible to the spread of infections, which requires only $3$ steps to infect a positive fraction of the whole graph. To prove this result, we rely on a quantitative lower bound for the maximum degree that might be of independent interest.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49196517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Joint convergence of sample cross-covariance matrices 样本交叉协方差矩阵的联合收敛性
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-03-22 DOI: 10.30757/alea.v20-14
M. Bhattacharjee, A. Bose, Apratim Dey
{"title":"Joint convergence of sample cross-covariance matrices","authors":"M. Bhattacharjee, A. Bose, Apratim Dey","doi":"10.30757/alea.v20-14","DOIUrl":"https://doi.org/10.30757/alea.v20-14","url":null,"abstract":"Suppose $X$ and $Y$ are $ptimes n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n to infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $rho$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, pto infty, p/nto yneq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $rho$. Independent copies of such matrices with same $p$ but different $n$, say ${n_l}$, different correlations ${rho_l}$, and different non-zero $y$'s, say ${y_l}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $sqrt{np^{-1}}(C-rho I_p)$ converges to an elliptic variable with parameter $rho^2$. In particular, this elliptic variable is circular when $rho=0$ and is semi-circular when $rho=1$. If we take independent $C_l$, then the matrices ${sqrt{n_lp^{-1}}(C_l-rho_l I_p)}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49108458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
The Spectra of Principal Submatrices in RotationallyInvariant Hermitian Random Matrices and the Markov–Krein Correspondence 旋转不变厄米随机矩阵中主子矩阵的谱及Markov-Krein对应
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-03-16 DOI: 10.30757/alea.v19-05
Katsunori Fujie, Takahiro Hasebe
{"title":"The Spectra of Principal Submatrices in Rotationally\u0000Invariant Hermitian Random Matrices and the Markov–\u0000Krein Correspondence","authors":"Katsunori Fujie, Takahiro Hasebe","doi":"10.30757/alea.v19-05","DOIUrl":"https://doi.org/10.30757/alea.v19-05","url":null,"abstract":"We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov–Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of m. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47553834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Cutoff for the Fredrickson-Andersen one spin facilitated model Fredrickson-Andersen单旋促进模型的截止点
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-02-26 DOI: 10.30757/alea.v19-20
Anatole Ertul
{"title":"Cutoff for the Fredrickson-Andersen one spin facilitated model","authors":"Anatole Ertul","doi":"10.30757/alea.v19-20","DOIUrl":"https://doi.org/10.30757/alea.v19-20","url":null,"abstract":"The Fredrickson-Andersen one spin facilitated model belongs to the class of Kinetically Constrained Spin Models. It is a non attractive process with positive spectral gap. In this paper we give a precise result on the relaxation for this process on an interval [1, L] starting from any initial configuration. A consequence of this result is that this process exhibits cutoff at time L/(2v) with window O( √ L) for a certain positive constant v. The key ingredient is the study of the evolution of the leftmost empty site in a filled infinite half-line called the front. In the process of the proof, we improve recent results about the front motion by showing that it evolves at speed v according to a uniform central limit theorem.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47548319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Factorization and discrete-time representation of multivariate CARMA processes 多元CARMA过程的因子分解和离散时间表示
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-02-23 DOI: 10.30757/alea.v19-31
Vicky Fasen-Hartmann, Markus Scholz
{"title":"Factorization and discrete-time representation of multivariate CARMA processes","authors":"Vicky Fasen-Hartmann, Markus Scholz","doi":"10.30757/alea.v19-31","DOIUrl":"https://doi.org/10.30757/alea.v19-31","url":null,"abstract":"In this paper we show that stationary and non-stationary multivariate continuous-time ARMA (MCARMA) processes have the representation as a sum of multivariate complexvalued Ornstein-Uhlenbeck processes under some mild assumptions. The proof benefits from properties of rational matrix polynomials. A conclusion is an alternative description of the autocovariance function of a stationary MCARMA process. Moreover, that representation is used to show that the discrete-time sampled MCARMA(p,q) process is a weak VARMA(p, p− 1) process if second moments exist. That result complements the weak VARMA(p, p− 1) representation derived in Chambers and Thornton [8]. In particular, it relates the right solvents of the autoregressive polynomial of the MCARMA process to the right solvents of the autoregressive polynomial of the VARMA process; in the one-dimensional case the right solvents are the zeros of the autoregressive polynomial. Finally, a factorization of the sample autocovariance function of the noise sequence is presented which is useful for statistical inference.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44652289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Refined Large Deviation Principle for Branching Brownian Motion Conditioned to Have a Low Maximum 具有低极大值条件下分支布朗运动的精细大偏差原理
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-02-18 DOI: 10.30757/alea.v19-34
Yanjia Bai, Lisa Hartung
{"title":"Refined Large Deviation Principle for Branching Brownian Motion Conditioned to Have a Low Maximum","authors":"Yanjia Bai, Lisa Hartung","doi":"10.30757/alea.v19-34","DOIUrl":"https://doi.org/10.30757/alea.v19-34","url":null,"abstract":"A BSTRACT . Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. In this note, we consider a branching Brownian motion conditioned to have a maximum below √ 2 α t ( α < 1). We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42374167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Tree convolution for probability distributionswith unbounded support 具有无界支持的概率分布的树卷积
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-02-01 DOI: 10.30757/alea.v18-58
E. Davis, David Jekel, Zhichao Wang
{"title":"Tree convolution for probability distributions\u0000with unbounded support","authors":"E. Davis, David Jekel, Zhichao Wang","doi":"10.30757/alea.v18-58","DOIUrl":"https://doi.org/10.30757/alea.v18-58","url":null,"abstract":"We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in\"An operad of non-commutative independences defined by trees\"(Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $boxplus_{mathcal{T}}(mu_1,dots,mu_N)$ for arbitrary probability measures $mu_1$, ..., $mu_N$ on $mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $mathcal{T}$-free convolution similar to Bercovici and Pata's results in the free case in\"Stable laws and domains of attraction in free probability\"(Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46169776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Covering a compact space by fixed-radius or growing random balls 用固定半径或生长随机球覆盖紧凑空间
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-01-29 DOI: 10.30757/alea.v19-29
D. Aldous
{"title":"Covering a compact space by fixed-radius or growing random balls","authors":"D. Aldous","doi":"10.30757/alea.v19-29","DOIUrl":"https://doi.org/10.30757/alea.v19-29","url":null,"abstract":". Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space S . In one specific model, “seeds\" arrive as a Poisson process (in time) at random positions with some distribution θ on S , and create balls whose radius increases at constant rate. By standardizing rates, the cover time C depends only on θ . The value χ ( S ) = min θ E θ C is a numerical characteristic of the compact space S , and we give weak general upper and lower bounds in terms of the covering numbers of S . This suggests a future research program of improving such general bounds, and estimating χ ( S ) for familiar examples of compact spaces. We treat one example, infinite product space [0 , 1] ∞ with the product topology. On a different theme, by analogy with the geometric models, and with the discrete coupon collector’s problem and with cover times for finite Markov chains, one expects a “weak concentration\" bound for the distribution of C to hold under minimal assumptions. We prove this as a simple consequence of a general result for increasing set-valued Markov processes.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47422473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Asymptotics for Kendall’s renewal function Kendall更新函数的渐近性
IF 0.7 4区 数学
Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-01-29 DOI: 10.30757/alea.v19-56
M. Cadena, B. Jasiulis-Gołdyn, E. Omey
{"title":"Asymptotics for Kendall’s renewal function","authors":"M. Cadena, B. Jasiulis-Gołdyn, E. Omey","doi":"10.30757/alea.v19-56","DOIUrl":"https://doi.org/10.30757/alea.v19-56","url":null,"abstract":". An elementary renewal theorem and a Blackwell theorem provided by Jasiulis-Gołdyn et al. (2020) in a setting of Kendall convolutions are proved under weaker hypothesis and extended to the Gamma class. Convergence rates of the limits concerned in these theorems are analyzed. Our theoretical results are illustrated by several examples involving novel probability distributions and extremes.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41741799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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