In this short note, we extend to the continuous case a mean projection theorem for discrete determinantal point processes associated with a finite range projection, thus strengthening a known result in random linear algebra due to Ermakov and Zolotukhin. We also give a new formula for the variance of the exterior power of the random projection.
{"title":"On the mean projection theorem for determinantal point processes","authors":"Adrien Kassel, Thierry L'evy","doi":"10.30757/ALEA.v20-17","DOIUrl":"https://doi.org/10.30757/ALEA.v20-17","url":null,"abstract":"In this short note, we extend to the continuous case a mean projection theorem for discrete determinantal point processes associated with a finite range projection, thus strengthening a known result in random linear algebra due to Ermakov and Zolotukhin. We also give a new formula for the variance of the exterior power of the random projection.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42166880","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}
For the renormalised sums of the random $pm 1$-colouring of the connected components of $mathbb Z$ generated by the coalescing renewal processes in the"power law P'olya's urn"of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz'alez Casanova, Kurt, and Span`o.
{"title":"Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn","authors":"Jan Lukas Igelbrink, A. Wakolbinger","doi":"10.30757/ALEA.v20-04","DOIUrl":"https://doi.org/10.30757/ALEA.v20-04","url":null,"abstract":"For the renormalised sums of the random $pm 1$-colouring of the connected components of $mathbb Z$ generated by the coalescing renewal processes in the\"power law P'olya's urn\"of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, Gonz'alez Casanova, Kurt, and Span`o.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43971488","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}
B. D. de Lima, S'ebastien Martineau, Humberto C. Sanna, D. Valesin
. Let L d = ( Z d , E d ) be the d -dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on L d in which every edge inside the s -dimensional sublattice Z s × { 0 } d − s , 2 ≤ s < d , is open with probability q and every other edge is open with probability p . We prove the uniqueness of the infinite cluster in the supercritical regime whenever p (cid:54) = p c ( d ) and 2 ≤ s < d − 1 , full uniqueness when s = d − 1 and that the critical point ( p, q c ( p )) can be approximated on the phase space by the critical points of slabs, for any p < p c ( d ) , where p c ( d ) denotes the threshold for homogeneous percolation.
。设L d = (Z d, E d)为d维超立方晶格。考虑L -d上的非齐次伯努利渗流模型，其中s维子格Z s × {0} d−s, 2≤s < d内的每条边都以概率q打开，其他每条边都以概率p打开。我们证明无限集群的独特性在超临界政权只要p (cid): 54) = p c (d)和2≤(s < d−1,全当s = d−1和独特性的临界点(p, q c (p))可以近似相空间临界点的石板,对于任何p < p c (d), p c (d)表示为均匀的渗滤阈值。
{"title":"Approximation on slabs and uniqueness for Bernoulli percolation with a sublattice of defects","authors":"B. D. de Lima, S'ebastien Martineau, Humberto C. Sanna, D. Valesin","doi":"10.30757/alea.v19-67","DOIUrl":"https://doi.org/10.30757/alea.v19-67","url":null,"abstract":". Let L d = ( Z d , E d ) be the d -dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on L d in which every edge inside the s -dimensional sublattice Z s × { 0 } d − s , 2 ≤ s < d , is open with probability q and every other edge is open with probability p . We prove the uniqueness of the infinite cluster in the supercritical regime whenever p (cid:54) = p c ( d ) and 2 ≤ s < d − 1 , full uniqueness when s = d − 1 and that the critical point ( p, q c ( p )) can be approximated on the phase space by the critical points of slabs, for any p < p c ( d ) , where p c ( d ) denotes the threshold for homogeneous percolation.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591443","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 provide a probabilistic representation of Lagrange’s identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for univariate target distribution under weak assumptions, in particular they hold for continuous and lattice distributions alike. The weights are studied under different sets of assumptions either on the test functions or on the underlying distributions. Many concrete illustrations for standard probability distributions are provided (including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy distributions).
{"title":"On Papathanasiou’s covariance expansions","authors":"Marie Ernst, G. Reinert, Yvik Swan","doi":"10.30757/alea.v19-69","DOIUrl":"https://doi.org/10.30757/alea.v19-69","url":null,"abstract":". In this paper we provide a probabilistic representation of Lagrange’s identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for univariate target distribution under weak assumptions, in particular they hold for continuous and lattice distributions alike. The weights are studied under different sets of assumptions either on the test functions or on the underlying distributions. Many concrete illustrations for standard probability distributions are provided (including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy distributions).","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591513","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 define joint exchangeability on arrays indexed by a vector of natural numbers with coordinates being the vertices of directed acyclic graphs (DAGs) using local isomorphisms. The notion provides a new version of exchangeability, which is a joint version of hierarchical exchangeability defined in Jung, L., Staton, Yang (2020). We also prove the existence of a generic representation by independent uniform random variables.
{"title":"A de Finetti-type representation of joint hierarchically exchangeable arrays on DAGs","authors":"Jiho Lee","doi":"10.30757/alea.v19-36","DOIUrl":"https://doi.org/10.30757/alea.v19-36","url":null,"abstract":". We define joint exchangeability on arrays indexed by a vector of natural numbers with coordinates being the vertices of directed acyclic graphs (DAGs) using local isomorphisms. The notion provides a new version of exchangeability, which is a joint version of hierarchical exchangeability defined in Jung, L., Staton, Yang (2020). We also prove the existence of a generic representation by independent uniform random variables.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591745","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 provide scaling limits for the block counting process and the fixation line of Λ coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0 , 1] satisfies (cid:82) [0 , 1] u − 1 | Λ − bλ | (d u ) < ∞ for some b ≥ 0 . Here λ denotes the Lebesgue measure on [0 , 1] . The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein–Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0 . We split the generators into two parts by additively decomposing Λ into a ‘Bolthausen–Sznitman part’ bλ and a ‘dust part’ Λ − bλ and then prove the uniform convergence of both parts separately.
{"title":"Scaling limits for the block counting process and the fixation line for a class of Λ-coalescents","authors":"M. Möhle, Benedict Vetter","doi":"10.30757/alea.v19-25","DOIUrl":"https://doi.org/10.30757/alea.v19-25","url":null,"abstract":". We provide scaling limits for the block counting process and the fixation line of Λ coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0 , 1] satisfies (cid:82) [0 , 1] u − 1 | Λ − bλ | (d u ) < ∞ for some b ≥ 0 . Here λ denotes the Lebesgue measure on [0 , 1] . The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein–Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0 . We split the generators into two parts by additively decomposing Λ into a ‘Bolthausen–Sznitman part’ bλ and a ‘dust part’ Λ − bλ and then prove the uniform convergence of both parts separately.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591493","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 study the simple random walk on the configuration model with given degree sequence (d1 , . . . , d n n) and investigate the connected components of its vacant set at level u > 0. We show that the size of the maximal connected component exhibits a phase transition at level u∗ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size n−1/3 around u∗ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph.
研究了给定度序列(d1，…)的构型模型上的简单随机漫步问题。， d nn)，研究其空集在水平u >0 0的连通分量。我们证明了最大连通分量的大小在u *水平上表现出一个相变，这个相变可以与某一Galton-Watson树上随机交错的临界参数有关。我们进一步证明了在u *周围存在一个大小为n−1/3的临界窗口，其中空集的最大连通分量具有类似于临界Erdős-Rényi随机图的度量空间缩放极限。
{"title":"Critical window for the vacant set left by random walk on the configuration model","authors":"J. Černý, T. Hayder","doi":"10.30757/alea.v19-10","DOIUrl":"https://doi.org/10.30757/alea.v19-10","url":null,"abstract":"We study the simple random walk on the configuration model with given degree sequence (d1 , . . . , d n n) and investigate the connected components of its vacant set at level u > 0. We show that the size of the maximal connected component exhibits a phase transition at level u∗ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size n−1/3 around u∗ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"66 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591154","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 study the two-sided level set of the two-dimensional discrete Gaussian free field (GFF), where a site is open if the absolute value of the GFF at this site is at most λ for a fixed parameter λ > 0 . For the GFF on a box of size N with Dirichlet boundary conditions, we show that there exists (cid:15) > 0 such that with probability tending to 1 as N → ∞ , all the open paths whose Euclidean diameters are of order N have lengths larger than N 1+ (cid:15) .
{"title":"On the chemical distance exponent for the two-sided level set of the two-dimensional Gaussian free field","authors":"Yifan Gao, Fuxi Zhang","doi":"10.30757/alea.v19-28","DOIUrl":"https://doi.org/10.30757/alea.v19-28","url":null,"abstract":". In this paper we study the two-sided level set of the two-dimensional discrete Gaussian free field (GFF), where a site is open if the absolute value of the GFF at this site is at most λ for a fixed parameter λ > 0 . For the GFF on a box of size N with Dirichlet boundary conditions, we show that there exists (cid:15) > 0 such that with probability tending to 1 as N → ∞ , all the open paths whose Euclidean diameters are of order N have lengths larger than N 1+ (cid:15) .","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591557","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 consider a “parabolic” scaling limit of tagged particle dynamics and that of empirical measure of the position of particles for stochastic ranking process with space-time dependent intensities. A stochastic ranking process is driven according to an algorithm for a self-organizing linear list of a finite number of items. We regard this process as a particle system. We fasten a tag to a “particle” (item) and observe the (normalized) motion of the “tagged particle”. We obtain a sum of diffusion processes between each two successive jump time for a “parabolic” scaling limit of tagged particle dynamics. In order to obtain the diffusion process, we have to observe a “parabolic” scaling limit of empirical measure of the position of particles. We also obtain a generalized Ornstein-Uhlenbeck process for a “parabolic” scaling limit of empirical measure of the position of particles.
{"title":"Functional central limit theorem for tagged particle dynamics in stochastic ranking process with space-time dependent intensities","authors":"Yukio Nagahata","doi":"10.30757/alea.v19-40","DOIUrl":"https://doi.org/10.30757/alea.v19-40","url":null,"abstract":". In this paper, we consider a “parabolic” scaling limit of tagged particle dynamics and that of empirical measure of the position of particles for stochastic ranking process with space-time dependent intensities. A stochastic ranking process is driven according to an algorithm for a self-organizing linear list of a finite number of items. We regard this process as a particle system. We fasten a tag to a “particle” (item) and observe the (normalized) motion of the “tagged particle”. We obtain a sum of diffusion processes between each two successive jump time for a “parabolic” scaling limit of tagged particle dynamics. In order to obtain the diffusion process, we have to observe a “parabolic” scaling limit of empirical measure of the position of particles. We also obtain a generalized Ornstein-Uhlenbeck process for a “parabolic” scaling limit of empirical measure of the position of particles.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591300","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 give a sequence of binary functions defined on the finite observations of a stationary point process which will almost surely eventually take the value POISSON if the observed process is Poisson
{"title":"A note on the Rényi criterion for Poisson processes and their identification","authors":"G. Morvai, B. Weiss","doi":"10.30757/alea.v19-66","DOIUrl":"https://doi.org/10.30757/alea.v19-66","url":null,"abstract":". We give a sequence of binary functions defined on the finite observations of a stationary point process which will almost surely eventually take the value POISSON if the observed process is Poisson","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591426","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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