Let { η i ( t ) , t ∈ [0 , 1] } ki =1 be independent copies of η = { η ( t ) , t ∈ [0 , 1] } , a mean zero continuous Gaussian process. Let This paper shows how exact local (at 0) and uniform moduli of continuity (on [0,1]) of Y k can be obtained from the exact local and uniform moduli of continuity of η .
{"title":"Local and uniform moduli of continuity of chi–square processes","authors":"M. Marcus, J. Rosen","doi":"10.1214/22-ecp471","DOIUrl":"https://doi.org/10.1214/22-ecp471","url":null,"abstract":"Let { η i ( t ) , t ∈ [0 , 1] } ki =1 be independent copies of η = { η ( t ) , t ∈ [0 , 1] } , a mean zero continuous Gaussian process. Let This paper shows how exact local (at 0) and uniform moduli of continuity (on [0,1]) of Y k can be obtained from the exact local and uniform moduli of continuity of η .","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42282025","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 short note, using the Kendall-Cranston coupling, we study on K"ahler (resp. quaternion K"ahler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.
{"title":"A note on first eigenvalue estimates by coupling methods in Kähler and quaternion Kähler manifolds","authors":"Fabrice Baudoin, Gunhee Cho, Guang Yang","doi":"10.1214/22-ecp452","DOIUrl":"https://doi.org/10.1214/22-ecp452","url":null,"abstract":"In this short note, using the Kendall-Cranston coupling, we study on K\"ahler (resp. quaternion K\"ahler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45522164","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}
Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${mathbb R}^n$. Given domains $U,W subseteq {mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning ${bf P}(T_U {bf P}(T_W t) > {bf P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.
{"title":"On the duration of stays of Brownian motion in domains in Euclidean space","authors":"Dimitrios Betsakos, Maher Boudabra, Greg Markowsky","doi":"10.1214/22-ecp498","DOIUrl":"https://doi.org/10.1214/22-ecp498","url":null,"abstract":"Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${mathbb R}^n$. Given domains $U,W subseteq {mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning ${bf P}(T_U {bf P}(T_W t) > {bf P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44842464","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 binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time 0 . The particle moves according to a Brownian motion with drift µ = 2 and diffusion coefficient σ 2 = 2 , until an independent exponential time of parameter 1 . At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to ∞ . The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.
{"title":"Total number of births on the negative half-line of the binary branching Brownian motion in the boundary case","authors":"Xinxing Chen, Bastien Mallein","doi":"10.1214/22-ecp449","DOIUrl":"https://doi.org/10.1214/22-ecp449","url":null,"abstract":"The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time 0 . The particle moves according to a Brownian motion with drift µ = 2 and diffusion coefficient σ 2 = 2 , until an independent exponential time of parameter 1 . At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to ∞ . The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"13 s1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41269457","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}
Anders Aamand, N. Alon, Jakob Bæk Tejs Knudsen, M. Thorup
We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,pi]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the trick of emph{exponential tilting}, we obtain sharp estimates for the point probabilities in the tail under a slightly stronger assumption on the random integer variables which we call strong monotonicity.
{"title":"On sums of monotone random integer variables","authors":"Anders Aamand, N. Alon, Jakob Bæk Tejs Knudsen, M. Thorup","doi":"10.1214/22-ecp500","DOIUrl":"https://doi.org/10.1214/22-ecp500","url":null,"abstract":"We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,pi]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the trick of emph{exponential tilting}, we obtain sharp estimates for the point probabilities in the tail under a slightly stronger assumption on the random integer variables which we call strong monotonicity.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41525666","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 that a random variable X of interest is observed perturbed by independent additive noise Y . This paper concerns the “the least favorable perturbation” ˆ Y ε , which maximizes the prediction error E ( X − E ( X | X + Y )) 2 in the class of Y with var ( Y ) ≤ ε . We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse.
{"title":"The least favorable noise","authors":"Philip A. Ernst, A. Kagan, L. Rogers","doi":"10.1214/22-ecp467","DOIUrl":"https://doi.org/10.1214/22-ecp467","url":null,"abstract":"Suppose that a random variable X of interest is observed perturbed by independent additive noise Y . This paper concerns the “the least favorable perturbation” ˆ Y ε , which maximizes the prediction error E ( X − E ( X | X + Y )) 2 in the class of Y with var ( Y ) ≤ ε . We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45012162","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}
By investigating McKean-Vlasov SDEs, the order preservation and positive correlation are characterized for nonlinear Fokker-Planck equations. The main results recover the corresponding criteria on these properties established in [3, 5] for diffusion processes or linear Fokker-Planck equations. AMS subject Classification: 60J60, 58J65.
{"title":"Order preservation and positive correlation for nonlinear Fokker-Planck equation","authors":"Panpan Ren","doi":"10.1214/22-ecp466","DOIUrl":"https://doi.org/10.1214/22-ecp466","url":null,"abstract":"By investigating McKean-Vlasov SDEs, the order preservation and positive correlation are characterized for nonlinear Fokker-Planck equations. The main results recover the corresponding criteria on these properties established in [3, 5] for diffusion processes or linear Fokker-Planck equations. AMS subject Classification: 60J60, 58J65.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43161389","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}
Let $M_n$ be a random $ntimes n$ matrix with i.i.d. $text{Bernoulli}(1/2)$ entries. We show that for fixed $kge 1$, [lim_{nto infty}frac{1}{n}log_2mathbb{P}[text{corank }M_nge k] = -k.]
{"title":"Rank deficiency of random matrices","authors":"Vishesh Jain, A. Sah, Mehtaab Sawhney","doi":"10.1214/22-ecp455","DOIUrl":"https://doi.org/10.1214/22-ecp455","url":null,"abstract":"Let $M_n$ be a random $ntimes n$ matrix with i.i.d. $text{Bernoulli}(1/2)$ entries. We show that for fixed $kge 1$, [lim_{nto infty}frac{1}{n}log_2mathbb{P}[text{corank }M_nge k] = -k.]","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42112126","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}
Is there a constant r0 such that, in any invariant tree network linking rate-1 Poisson points in the plane, the mean within-network distance between points at Euclidean distance r is infinite for r>r0? We prove a slightly weaker result. This is a continuum analog of a result of Benjamini et al (2001) on invariant spanning trees of the integer lattice.
{"title":"Route lengths in invariant spatial tree networks","authors":"D. Aldous","doi":"10.1214/21-ECP401","DOIUrl":"https://doi.org/10.1214/21-ECP401","url":null,"abstract":"Is there a constant r0 such that, in any invariant tree network linking rate-1 Poisson points in the plane, the mean within-network distance between points at Euclidean distance r is infinite for r>r0? We prove a slightly weaker result. This is a continuum analog of a result of Benjamini et al (2001) on invariant spanning trees of the integer lattice.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"26 1","pages":"1-12"},"PeriodicalIF":0.5,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45945655","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 spectrum of the kinetic Brownian motion in the space of d × d Hermitian matrices, d ≥ 2 . We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair (Λ , ˙Λ) of Λ and its derivative is Markovian) if and only if d = 2 . In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.
{"title":"Kinetic Dyson Brownian motion","authors":"P. Perruchaud","doi":"10.1214/22-ecp480","DOIUrl":"https://doi.org/10.1214/22-ecp480","url":null,"abstract":"We study the spectrum of the kinetic Brownian motion in the space of d × d Hermitian matrices, d ≥ 2 . We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair (Λ , ˙Λ) of Λ and its derivative is Markovian) if and only if d = 2 . In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46974265","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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