C. Hillairet, Lorick Huang, Mahmoud Khabou, Anthony Reveillac
. 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
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
我们发展了第二作者和刘卫华在《树定义的非交换独立算子》（数学论文，2020，doi:10.4064/dm797-6-2020）中研究的树卷积的复分析观点，推广了自由卷积、布尔卷积、单调卷积和正交卷积。特别地，对于$N$正则树的每个根子树$mathcal｛T｝$（顶点由交替字符串标记），我们定义了任意概率测度$ma_1$，…的卷积$boxplus_｛mathcal{T｝｝（mu_1，dots，mu_N）$$mu_N$在$mathbb｛R｝$上使用Cauchy变换的特定定点方程。卷积运算遵循doi:10.4064/dm797-6-2020中的树运算器的运算器结构。我们证明了迭代$mathcal{T}$自由卷积的一个一般极限定理，类似于Bercovici和Pata在“自由概率中的稳定定律和吸引域”（Annals of Mathematics，1999，doi:10.2307/12180）中在自由情况下的结果，并推导出每个经典稳定定律在吸引域中的测度的极限定理。
{"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}
. 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.
． 简单的随机覆盖模型，在欧氏空间中得到了很好的研究，也可以在一般紧度量空间S上定义。在一个特定的模型中，“种子”作为泊松过程(在时间上)到达随机位置，在S上有一些分布θ，并产生半径以恒定速率增加的球。通过标准化速率，覆盖时间C只取决于θ。值χ (S) = min θ E θ C是紧空间S的一个数值特征，并给出了S的覆盖数的弱一般上界和下界。这暗示了一个未来的研究计划，改进这样的一般界限，并估计χ (S)为熟悉的紧空间的例子。我们处理一个例子，无限积空间[0,1]∞与积拓扑。在另一个主题上，通过类比几何模型、离散券集问题和有限马尔可夫链的覆盖时间，我们期望C分布的“弱集中”界在最小假设下成立。我们用一个关于集值递增马尔可夫过程的一般结果的一个简单结论来证明这一点。
{"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}
. 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.
{"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}
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