Aizelle Abelgas, Bryan Carrillo, John Palacios, David Weisbart, Adam M. Yassine
A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.
{"title":"Buffon’s problem determines Gaussian curvature in three geometries","authors":"Aizelle Abelgas, Bryan Carrillo, John Palacios, David Weisbart, Adam M. Yassine","doi":"10.1017/jpr.2023.114","DOIUrl":"https://doi.org/10.1017/jpr.2023.114","url":null,"abstract":"A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583945","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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline1.png" /> <jats:tex-math> $mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the family of all coherent distributions on the unit square <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline2.png" /> <jats:tex-math> $[0,1]^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, i.e. all those probability measures <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline3.png" /> <jats:tex-math> $mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which there exists a random vector <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline4.png" /> <jats:tex-math> $(X,Y)sim mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, a pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline5.png" /> <jats:tex-math> $(mathcal{G},mathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline6.png" /> <jats:tex-math> $sigma$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-fields, and an event <jats:italic>E</jats:italic> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline7.png" /> <jats:tex-math> $X=mathbb{P}(Emidmathcal{G})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline8.png" /> <jats:tex-math> $Y=mathbb{P}(Emidmathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> almost surely. We examine the set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline9.png" /> <jats:tex-math> $mathrm{ext}(mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of extreme points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900224000019_inline10.png" /> <jats:tex-math> $mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and provide its general characterisation. Moreover, we establish sever
{"title":"Coherent distributions on the square–extreme points and asymptotics","authors":"Stanisław Cichomski, Adam Osękowski","doi":"10.1017/jpr.2024.1","DOIUrl":"https://doi.org/10.1017/jpr.2024.1","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline1.png\" /> <jats:tex-math> $mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the family of all coherent distributions on the unit square <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline2.png\" /> <jats:tex-math> $[0,1]^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, i.e. all those probability measures <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline3.png\" /> <jats:tex-math> $mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which there exists a random vector <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline4.png\" /> <jats:tex-math> $(X,Y)sim mu$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, a pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline5.png\" /> <jats:tex-math> $(mathcal{G},mathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline6.png\" /> <jats:tex-math> $sigma$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-fields, and an event <jats:italic>E</jats:italic> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline7.png\" /> <jats:tex-math> $X=mathbb{P}(Emidmathcal{G})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline8.png\" /> <jats:tex-math> $Y=mathbb{P}(Emidmathcal{H})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> almost surely. We examine the set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline9.png\" /> <jats:tex-math> $mathrm{ext}(mathcal{C})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of extreme points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000019_inline10.png\" /> <jats:tex-math> $mathcal{C}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and provide its general characterisation. Moreover, we establish sever","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"53 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140584029","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}
Ze-An Ng, You-Beng Koh, Tee-How Loo, Hailiang Yang
We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option.
{"title":"Super-replication of life-contingent options under the Black–Scholes framework","authors":"Ze-An Ng, You-Beng Koh, Tee-How Loo, Hailiang Yang","doi":"10.1017/jpr.2024.10","DOIUrl":"https://doi.org/10.1017/jpr.2024.10","url":null,"abstract":"We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140584023","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 investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times, and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type and to derive expressions for the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors.
{"title":"Constrained optimal stopping under a regime-switching model","authors":"Takuji Arai, Masahiko Takenaka","doi":"10.1017/jpr.2023.122","DOIUrl":"https://doi.org/10.1017/jpr.2023.122","url":null,"abstract":"<p>We investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times, and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type and to derive expressions for the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"25 3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302726","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 show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.
{"title":"Inequalities between time and customer averages for HNB(W)UE arrival processes","authors":"Shigeo Shioda, Kana Nakano","doi":"10.1017/jpr.2023.120","DOIUrl":"https://doi.org/10.1017/jpr.2023.120","url":null,"abstract":"We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140200586","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}
Rita Giuliano, Claudio Macci, Barbara Pacchiarotti
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed exponential random variables.
{"title":"Asymptotic results for sums and extremes","authors":"Rita Giuliano, Claudio Macci, Barbara Pacchiarotti","doi":"10.1017/jpr.2023.118","DOIUrl":"https://doi.org/10.1017/jpr.2023.118","url":null,"abstract":"<p>The term <span>moderate deviations</span> is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant, and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about <span>noncentral moderate deviations</span> when the weak convergence is towards a non-Gaussian distribution. In this paper we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of independent and identically distributed random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of independent and identically distributed <span>exponential</span> random variables.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"39 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140115903","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}
Panna Tímea Fekete, Roland Molontay, Balázs Ráth, Kitti Varga
We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.
{"title":"Color-avoiding percolation and branching processes","authors":"Panna Tímea Fekete, Roland Molontay, Balázs Ráth, Kitti Varga","doi":"10.1017/jpr.2023.115","DOIUrl":"https://doi.org/10.1017/jpr.2023.115","url":null,"abstract":"We study a variant of the color-avoiding percolation model introduced by Krause <jats:italic>et al.</jats:italic>, namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075069","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 a skew Ornstein–Uhlenbeck process with zero being a sticky reflecting boundary, which is defined as the weak solution to a stochastic differential equation (SDE) system involving local time. The main results obtained include: (i) the existence and uniqueness of solutions to the SDE system, (ii) the scale function and speed measure, and (iii) the distributional properties regarding the transition density and the first hitting times. On the application side, we apply the process to interest rate modeling and obtain the explicit pricing formula for zero-coupon bonds. Numerical examples illustrate the impacts on bond yields of skewness and stickiness parameters.
{"title":"Skew Ornstein–Uhlenbeck processes with sticky reflection and their applications to bond pricing","authors":"Shiyu Song, Guangli Xu","doi":"10.1017/jpr.2023.110","DOIUrl":"https://doi.org/10.1017/jpr.2023.110","url":null,"abstract":"<p>We study a skew Ornstein–Uhlenbeck process with zero being a sticky reflecting boundary, which is defined as the weak solution to a stochastic differential equation (SDE) system involving local time. The main results obtained include: (i) the existence and uniqueness of solutions to the SDE system, (ii) the scale function and speed measure, and (iii) the distributional properties regarding the transition density and the first hitting times. On the application side, we apply the process to interest rate modeling and obtain the explicit pricing formula for zero-coupon bonds. Numerical examples illustrate the impacts on bond yields of skewness and stickiness parameters.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043941","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 asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $nptoinfty$ and $n^2(1-p)toinfty$ .
{"title":"Average Jaccard index of random graphs","authors":"Qunqiang Feng, Shuai Guo, Zhishui Hu","doi":"10.1017/jpr.2023.112","DOIUrl":"https://doi.org/10.1017/jpr.2023.112","url":null,"abstract":"The asymptotic behavior of the Jaccard index in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>), the classical Erdös–Rényi random graph model, is studied as <jats:italic>n</jats:italic> goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>) is shown to be asymptotically normal under an additional mild condition that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223001122_inline1.png\" /> <jats:tex-math> $nptoinfty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223001122_inline2.png\" /> <jats:tex-math> $n^2(1-p)toinfty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979356","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：