Pub Date : 2024-03-24DOI: 10.3103/s0027132223060050
V. I. Piterbarg
Abstract
The paper studies the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu.V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A.N. Kolmogorov proposed this problem in the early 1950s.
{"title":"Asymptotic Behavior of Point Processes of Exceeding the High Levels of Gaussian Stationary Sequence","authors":"V. I. Piterbarg","doi":"10.3103/s0027132223060050","DOIUrl":"https://doi.org/10.3103/s0027132223060050","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper studies the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu.V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A.N. Kolmogorov proposed this problem in the early 1950s.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"455 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-24DOI: 10.3103/s0027132223060086
A. N. Vetokhin
Abstract
We consider a parametric family of dynamical systems defined on a locally compact metric space and continuously dependent on a parameter from some metric space. For any such family, the local entropy of the dynamical systems included in it is studied as a function of a parameter from the point of view of the Baire classification of functions.
{"title":"Sharp Baire Classification of Local Entropy of Parametric Families of Dynamical Systems","authors":"A. N. Vetokhin","doi":"10.3103/s0027132223060086","DOIUrl":"https://doi.org/10.3103/s0027132223060086","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider a parametric family of dynamical systems defined on a locally compact metric space and continuously dependent on a parameter from some metric space. For any such family, the local entropy of the dynamical systems included in it is studied as a function of a parameter from the point of view of the Baire classification of functions.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"104 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-24DOI: 10.3103/s0027132223060062
O. S. Rozanova
Abstract
The paper considers Kolmogorov–Feller equation for the probability density of a Markov process on a half-axis, which arises in important problems of biology. This process consists of random jumps distributed according to the Laplace law and a deterministic return to zero. It is shown that the Green function for such an equation can be found both in the form of a series and in explicit form for some ratios of the parameters. This allows finding explicit solutions to the Kolmogorov–Feller equation for many initial data.
{"title":"On the Solution to the Kolmogorov-Feller Equation Arising in a Biological Evolution Model","authors":"O. S. Rozanova","doi":"10.3103/s0027132223060062","DOIUrl":"https://doi.org/10.3103/s0027132223060062","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper considers Kolmogorov–Feller equation for the probability density of a Markov process on a half-axis, which arises in important problems of biology. This process consists of random jumps distributed according to the Laplace law and a deterministic return to zero. It is shown that the Green function for such an equation can be found both in the form of a series and in explicit form for some ratios of the parameters. This allows finding explicit solutions to the Kolmogorov–Feller equation for many initial data.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"16 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-24DOI: 10.3103/s0027132223060074
M. P. Savelov
Abstract
Let (X_{1},X_{2},ldots,X_{n}) be independent identically distributed random vectors. Consider a vector (V(X_{1},X_{2},ldots,X_{n})) whose each component is either a (U)-statistic or an (M)-estimator. Sufficient conditions for asymptotic normality of the vector (V(X_{1},X_{2},ldots,X_{n})) are obtained. In the case when (X_{1},X_{2},ldots) are one-dimensional, sufficient conditions for asymptotic normality are obtained for a vector, each component of which is either a (U)-statistic, or an (M)-estimator, or a sample quantile.
AbstractLet (X_{1},X_{2},ldots,X_{n}) are independent identically distributed random vector.考虑一个向量 (V(X_{1},X_{2},ldots,X_{n}))，它的每个分量都是(U)-统计量或(M)-估计量。得到了向量 (V(X_{1},X_{2},ldots,X_{n}))渐近正态性的充分条件。在(X_{1},X_{2},ldots)是一维的情况下，得到了向量渐近正态性的充分条件，该向量的每个分量要么是一个(U)-统计量，要么是一个(M)-估计量，要么是一个样本量位数。
{"title":"Limit Joint Distribution of $$boldsymbol{U}$$ -Statistics, $$boldsymbol{M}$$ -Estimates, and Sample Quantiles","authors":"M. P. Savelov","doi":"10.3103/s0027132223060074","DOIUrl":"https://doi.org/10.3103/s0027132223060074","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Let <span>(X_{1},X_{2},ldots,X_{n})</span> be independent identically distributed random vectors. Consider a vector <span>(V(X_{1},X_{2},ldots,X_{n}))</span> whose each component is either a <span>(U)</span>-statistic or an <span>(M)</span>-estimator. Sufficient conditions for asymptotic normality of the vector <span>(V(X_{1},X_{2},ldots,X_{n}))</span> are obtained. In the case when <span>(X_{1},X_{2},ldots)</span> are one-dimensional, sufficient conditions for asymptotic normality are obtained for a vector, each component of which is either a <span>(U)</span>-statistic, or an <span>(M)</span>-estimator, or a sample quantile.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"32 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-24DOI: 10.3103/s0027132223060025
E. V. Bulinskaya
Abstract
Two discrete-time insurance models are considered. The first model studies nonproportional reinsurance and bank loans. For this model, we establish the optimal control and stability to small fluctuation of parameters and perturbation of random variables distributions describing the model. The second model is dual and the ruin probabilities are compared under assumption that the gains distributions satisfy one of four partial orders.
{"title":"Discrete-Time Insurance Models","authors":"E. V. Bulinskaya","doi":"10.3103/s0027132223060025","DOIUrl":"https://doi.org/10.3103/s0027132223060025","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Two discrete-time insurance models are considered. The first model studies nonproportional reinsurance and bank loans. For this model, we establish the optimal control and stability to small fluctuation of parameters and perturbation of random variables distributions describing the model. The second model is dual and the ruin probabilities are compared under assumption that the gains distributions satisfy one of four partial orders.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"31 2 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3103/s0027132223050029
M. V. Boldin, A. R. Shabakaeva
Abstract
We consider a stationary linear (AR(p)) model with zero mean. The autoregression parameters, as well as the distribution function (d.f.) (G(x)) of innovations, are unknown. We test symmetry of innovations with respect to zero in two situations. In the first case the observations are a sample from a stationary solution of (AR(p)). We estimate parameters and find residuals. Based on them we construct a kind of empirical d.f. and the omega-square type test statistic. Its asymptotic d.f. under the hypothesis and the local alternatives are found. In the second situation the observations are subject to gross errors (outliers). For testing the symmetry of innovations we again construct the Pearson’s type statistic and find its asymptotic d.f. under the hypothesis and the local alternatives. We establish the asymptotic robustness of Pearson’s test as well.
Abstract We consider a stationary linear (AR(p)) model with zero mean.自回归参数以及创新值的分布函数（d.f. ）(G(x))都是未知的。我们在两种情况下检验创新值相对于零的对称性。在第一种情况下，观测值是从 (AR(p)) 的静态解中抽取的样本。我们估计参数并找到残差。在此基础上，我们构建了一种经验 d.f.和Ω-平方类型检验统计量。在假设和局部替代条件下，找到其渐近 d.f.。在第二种情况下，观测数据会出现严重错误（异常值）。为了检验创新的对称性，我们再次构建皮尔逊类型统计量，并找出其在假设和局部替代方案下的渐近 d.f.。我们还建立了皮尔逊检验的渐近稳健性。
{"title":"On Testing the Symmetry of Innovation Distribution in Autoregression Schemes","authors":"M. V. Boldin, A. R. Shabakaeva","doi":"10.3103/s0027132223050029","DOIUrl":"https://doi.org/10.3103/s0027132223050029","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider a stationary linear <span>(AR(p))</span> model with zero mean. The autoregression parameters, as well as the distribution function (d.f.) <span>(G(x))</span> of innovations, are unknown. We test symmetry of innovations with respect to zero in two situations. In the first case the observations are a sample from a stationary solution of <span>(AR(p))</span>. We estimate parameters and find residuals. Based on them we construct a kind of empirical d.f. and the omega-square type test statistic. Its asymptotic d.f. under the hypothesis and the local alternatives are found. In the second situation the observations are subject to gross errors (outliers). For testing the symmetry of innovations we again construct the Pearson’s type statistic and find its asymptotic d.f. under the hypothesis and the local alternatives. We establish the asymptotic robustness of Pearson’s test as well.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"13 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138742965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3103/s0027132223050078
D. I. Vasilev, E. E. Gasanov
Abstract
The paper considers the application of the locator cellular automaton model to the closest neighbor search problem. The locator cellular automaton model assumes the possibility for each cell to translate a signal through any distance using the ether. It was proven earlier that the ether model allows solving the problem with logarithmic time. In this paper we have derived a logarithmic lower bound for this problem.
{"title":"A Lower Bound on Complexity of a Locator Cellular Automaton Solution for the Closest Neighbor Search Problem","authors":"D. I. Vasilev, E. E. Gasanov","doi":"10.3103/s0027132223050078","DOIUrl":"https://doi.org/10.3103/s0027132223050078","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper considers the application of the locator cellular automaton model to the closest neighbor search problem. The locator cellular automaton model assumes the possibility for each cell to translate a signal through any distance using the ether. It was proven earlier that the ether model allows solving the problem with logarithmic time. In this paper we have derived a logarithmic lower bound for this problem.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"3 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138743051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3103/s0027132223050030
A. A. Kuznetsova
Abstract
Examples of degenerate (non-Morse-type) multisaddle singularities (atoms) of complexity 1 and multiplicity 3 are realized by integrable billiard books.
摘要通过可积分台球书实现了复杂度为 1 和乘数为 3 的退化（非莫氏型）多鞍奇异点（原子）的实例。
{"title":"Modeling the Degenerate Singularities of Integrable Billiard Systems by Billiard Books","authors":"A. A. Kuznetsova","doi":"10.3103/s0027132223050030","DOIUrl":"https://doi.org/10.3103/s0027132223050030","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Examples of degenerate (non-Morse-type) multisaddle singularities (atoms) of complexity 1 and multiplicity 3 are realized by integrable billiard books.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"238 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138742897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3103/s0027132223050042
M. A. Nikulin
Abstract
The stationary Schrödinger equation is studied in a domain bounded by two confocal ellipses and in its coverings. The order of dependence of the Laplace operator eigenvalues on sufficiently small distance between the foci is obtained. Coefficients of the power series expansion of said eigenvalues are calculated up to and including the square of half the focal distance.
{"title":"Spectrum of Schrödinger Operator in Covering of Elliptic Ring","authors":"M. A. Nikulin","doi":"10.3103/s0027132223050042","DOIUrl":"https://doi.org/10.3103/s0027132223050042","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The stationary Schrödinger equation is studied in a domain bounded by two confocal ellipses and in its coverings. The order of dependence of the Laplace operator eigenvalues on sufficiently small distance between the foci is obtained. Coefficients of the power series expansion of said eigenvalues are calculated up to and including the square of half the focal distance.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"13 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138742798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3103/s0027132223050066
N. F. Valeev, E. A. Nazirova, Ya. T. Sultanaev
Abstract
The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm–Liouville equation.
{"title":"Constructing the Asymptotics of Solutions to Differential Sturm–Liouville Equations in Classes of Oscillating Coefficients","authors":"N. F. Valeev, E. A. Nazirova, Ya. T. Sultanaev","doi":"10.3103/s0027132223050066","DOIUrl":"https://doi.org/10.3103/s0027132223050066","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm–Liouville equation.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138742842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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