This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.
{"title":"Introducing smooth amnesia to the memory of the Elephant Random Walk","authors":"Lucile Laulin","doi":"10.1214/22-ecp495","DOIUrl":"https://doi.org/10.1214/22-ecp495","url":null,"abstract":"This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42606328","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 extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble (cid:80) i (cid:54) = j f ( L N ( θ i − θ j )) for arbitrary β > 0 in the mesoscopic regime L N = O ( N 2 / 3 − (cid:15) ) . In addition, we obtain similar results for bipartite statistics.
{"title":"Central limit theorem for CβE pair dependent statistics in mesoscopic regime","authors":"A. Aguirre, A. Soshnikov","doi":"10.1214/22-ecp481","DOIUrl":"https://doi.org/10.1214/22-ecp481","url":null,"abstract":"We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble (cid:80) i (cid:54) = j f ( L N ( θ i − θ j )) for arbitrary β > 0 in the mesoscopic regime L N = O ( N 2 / 3 − (cid:15) ) . In addition, we obtain similar results for bipartite statistics.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43971099","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}
{"title":"Simple halfspace depth","authors":"P. Laketa, D. Pokorný, Stanislav Nagy","doi":"10.1214/22-ecp503","DOIUrl":"https://doi.org/10.1214/22-ecp503","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48019149","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}
To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.
{"title":"Distribution dependent SDEs for Navier-Stokes type equations","authors":"Fengchao Wang","doi":"10.1214/22-ecp479","DOIUrl":"https://doi.org/10.1214/22-ecp479","url":null,"abstract":"To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44821491","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}
Consider a 2-dimensional soft random geometric graph G ( λ, s, φ ), obtained by placing a Poisson( λs 2 ) number of vertices uniformly at random in a square of side s , with edges placed between each pair x, y of vertices with probability φ ( (cid:107) x − y (cid:107) ), where φ : R + → [0 , 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G ( λ, s, φ ) in the large- s limit with ( λ, φ ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λ c ( φ ).
{"title":"Giant component of the soft random geometric graph","authors":"M. Penrose","doi":"10.1214/22-ecp491","DOIUrl":"https://doi.org/10.1214/22-ecp491","url":null,"abstract":"Consider a 2-dimensional soft random geometric graph G ( λ, s, φ ), obtained by placing a Poisson( λs 2 ) number of vertices uniformly at random in a square of side s , with edges placed between each pair x, y of vertices with probability φ ( (cid:107) x − y (cid:107) ), where φ : R + → [0 , 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G ( λ, s, φ ) in the large- s limit with ( λ, φ ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λ c ( φ ).","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42670191","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 existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.
{"title":"McKean-Vlasov equations with positive feedback through elastic stopping times","authors":"B. Hambly, Julian Meier","doi":"10.1214/22-ecp482","DOIUrl":"https://doi.org/10.1214/22-ecp482","url":null,"abstract":"We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48157972","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 analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.
{"title":"A random graph of moderate density","authors":"Á. Backhausz, T. F. Móri","doi":"10.1214/21-ecp444","DOIUrl":"https://doi.org/10.1214/21-ecp444","url":null,"abstract":"We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n 1 / 6 ). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46783625","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 large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.
{"title":"Large-time and small-time behaviors of the spectral heat content for time-changed stable processes","authors":"Kei Kobayashi, Hyunchul Park","doi":"10.1214/22-ecp478","DOIUrl":"https://doi.org/10.1214/22-ecp478","url":null,"abstract":"We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47368804","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 note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension [18] of the classical framework of Toom [20]. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions [1, 2], but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.
{"title":"Subcritical bootstrap percolation via Toom contours","authors":"Ivailo Hartarsky, R. Szab'o","doi":"10.1214/22-ecp496","DOIUrl":"https://doi.org/10.1214/22-ecp496","url":null,"abstract":"In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension [18] of the classical framework of Toom [20]. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions [1, 2], but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43100680","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 note, we consider the parabolic Anderson model on R + × R , driven by a Gaussian noise which is fractional in time with index H 0 > 1 / 2 and fractional in space with index 0 < H < 1 / 2 such that H 0 + H > 3 / 4. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p -th moments with p ≥ 2.
{"title":"Parabolic Anderson model with rough noise in space and rough initial conditions","authors":"R. Balan, Le Chen, Yiping Ma","doi":"10.1214/22-ecp506","DOIUrl":"https://doi.org/10.1214/22-ecp506","url":null,"abstract":"In this note, we consider the parabolic Anderson model on R + × R , driven by a Gaussian noise which is fractional in time with index H 0 > 1 / 2 and fractional in space with index 0 < H < 1 / 2 such that H 0 + H > 3 / 4. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p -th moments with p ≥ 2.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49497306","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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