Suppose that a random variable $X$ of interest is observed. This paper concerns"the least favorable noise"$hat{Y}_{epsilon}$, which maximizes the prediction error $E [X - E[X|X+Y]]^2 $ (or minimizes the variance of $E[X| X+Y]$) in the class of $Y$ with $Y$ independent of $X$ and $mathrm{var} Y leq epsilon^2$. This problem was first studied by Ernst, Kagan, and Rogers ([3]). In the present manuscript, we show that the least favorable noise $hat{Y}_{epsilon}$ must exist and that its variance must be $epsilon^2$. The proof of existence relies on a convergence result we develop for variances of conditional expectations. Further, we show that the function $inf_{mathrm{var} Y leq epsilon^2} , mathrm{var} , E[X|X+Y]$ is both strictly decreasing and right continuous in $epsilon$.
假设观察到一个感兴趣的随机变量$X$。本文关注的是“最不利噪声”$hat{Y}_{epsilon}$，它在$Y$类中使预测误差$E [X - E[X|X+Y]]^2 $最大化(或使方差$E[X| X+Y]$最小化)，而$Y$独立于$X$和$mathrm{var} Y leq epsilon^2$。这个问题最早是由恩斯特、卡根和罗杰斯(b[3])研究的。在本文中，我们证明了最不利噪声$hat{Y}_{epsilon}$必须存在，其方差必须为$epsilon^2$。存在性的证明依赖于我们为条件期望的方差开发的收敛结果。进一步，我们证明了函数$inf_{mathrm{var} Y leq epsilon^2} , mathrm{var} , E[X|X+Y]$在$epsilon$中既是严格递减的又是右连续的。
{"title":"The existence of the least favorable noise","authors":"Dongzhou Huang","doi":"10.1214/23-ecp533","DOIUrl":"https://doi.org/10.1214/23-ecp533","url":null,"abstract":"Suppose that a random variable $X$ of interest is observed. This paper concerns\"the least favorable noise\"$hat{Y}_{epsilon}$, which maximizes the prediction error $E [X - E[X|X+Y]]^2 $ (or minimizes the variance of $E[X| X+Y]$) in the class of $Y$ with $Y$ independent of $X$ and $mathrm{var} Y leq epsilon^2$. This problem was first studied by Ernst, Kagan, and Rogers ([3]). In the present manuscript, we show that the least favorable noise $hat{Y}_{epsilon}$ must exist and that its variance must be $epsilon^2$. The proof of existence relies on a convergence result we develop for variances of conditional expectations. Further, we show that the function $inf_{mathrm{var} Y leq epsilon^2} , mathrm{var} , E[X|X+Y]$ is both strictly decreasing and right continuous in $epsilon$.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47464978","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":"Corrigendum to: The contact process on periodic trees","authors":"Xiangying Huang, R. Durrett","doi":"10.1214/23-ecp518","DOIUrl":"https://doi.org/10.1214/23-ecp518","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44393029","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 a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [S. Junk: Communications in Mathematical Physics 389, 1087–1097 (2022)]. We extend this characterization to a large class of unbounded environments which includes many commonly used distributions.
{"title":"Moment characterization of the weak disorder phase for directed polymers in a class of unbounded environments","authors":"Ryoki Fukushima, Stefan Junk","doi":"10.1214/23-ecp545","DOIUrl":"https://doi.org/10.1214/23-ecp545","url":null,"abstract":"For a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [S. Junk: Communications in Mathematical Physics 389, 1087–1097 (2022)]. We extend this characterization to a large class of unbounded environments which includes many commonly used distributions.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135958441","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 that the scaling limit of the weakly self-avoiding walk on a d-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is o(V1∕2) where V is the volume (number of points) of the torus and if d>4. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on Zd. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the torus up until it reaches about V1∕2 steps, which we believe is sharp.
{"title":"The scaling limit of the weakly self-avoiding walk on a high-dimensional torus","authors":"Emmanuel Michta","doi":"10.1214/23-ecp531","DOIUrl":"https://doi.org/10.1214/23-ecp531","url":null,"abstract":"We prove that the scaling limit of the weakly self-avoiding walk on a d-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is o(V1∕2) where V is the volume (number of points) of the torus and if d>4. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on Zd. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the torus up until it reaches about V1∕2 steps, which we believe is sharp.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135733740","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}
Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on Rd with intensity λ>0 is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate α>0, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of λ and α, the infection almost surely dies out.
{"title":"Brownian snails with removal die out in one dimension","authors":"Ivailo Hartarsky, Lyuben Lichev","doi":"10.1214/23-ecp551","DOIUrl":"https://doi.org/10.1214/23-ecp551","url":null,"abstract":"Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on Rd with intensity λ>0 is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate α>0, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of λ and α, the infection almost surely dies out.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135450721","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 investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are able to construct a sequence of infinitely differentiable functions having the same Lipschitz constant as the original function. This solves an open problem raised in [11]. For (resp. twice) continuously differentiable function, we show that our approximation also holds for the first-order derivative (resp. second-order derivatives), therefore solving another open problem raised in [11].
{"title":"On smooth approximations in the Wasserstein space","authors":"Andrea Cosso, Mattia Martini","doi":"10.1214/23-ECP538","DOIUrl":"https://doi.org/10.1214/23-ECP538","url":null,"abstract":"In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are able to construct a sequence of infinitely differentiable functions having the same Lipschitz constant as the original function. This solves an open problem raised in [11]. For (resp. twice) continuously differentiable function, we show that our approximation also holds for the first-order derivative (resp. second-order derivatives), therefore solving another open problem raised in [11].","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45296634","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":"Generating discrete uniform distribution from a biased coin using number-theoretic method","authors":"Xiaoyu Lei","doi":"10.1214/23-ecp537","DOIUrl":"https://doi.org/10.1214/23-ecp537","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41951614","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 specific relative entropy, introduced in the Wiener space setting by N. Gantert, allows to quantify the discrepancy between the laws of potentially mutually singular measures. It appears naturally as the large deviations rate function in a randomized version of Donsker’s invariance principle, as well as in a novel transport-information inequality recently derived by H. Föllmer. A conjecture, put forward by the aforementioned authors, concerns a closed form expression for the specific relative entropy between continuous martingale laws in terms of their quadratic variations. We provide a first partial result in this direction, by establishing this conjecture in the case of well-behaved martingale diffusions on the line.
{"title":"On the specific relative entropy between martingale diffusions on the line","authors":"Julio Backhoff-Veraguas, Clara Unterberger","doi":"10.1214/23-ecp548","DOIUrl":"https://doi.org/10.1214/23-ecp548","url":null,"abstract":"The specific relative entropy, introduced in the Wiener space setting by N. Gantert, allows to quantify the discrepancy between the laws of potentially mutually singular measures. It appears naturally as the large deviations rate function in a randomized version of Donsker’s invariance principle, as well as in a novel transport-information inequality recently derived by H. Föllmer. A conjecture, put forward by the aforementioned authors, concerns a closed form expression for the specific relative entropy between continuous martingale laws in terms of their quadratic variations. We provide a first partial result in this direction, by establishing this conjecture in the case of well-behaved martingale diffusions on the line.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135955011","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":"Competition on Zd driven by branching random walk","authors":"Maria Deijfen, Timo Vilkas","doi":"10.1214/23-ecp521","DOIUrl":"https://doi.org/10.1214/23-ecp521","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49209909","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 existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.
{"title":"On the existence of weak solutions to stochastic Volterra equations","authors":"David J. Prömel, David Scheffels","doi":"10.1214/23-ecp554","DOIUrl":"https://doi.org/10.1214/23-ecp554","url":null,"abstract":"The existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704896","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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