Pub Date : 2025-08-18DOI: 10.1016/j.spl.2025.110529
Jinjin Zhang , Yang Yang , Lin Xu
Consider a bidimensional compound risk model with stochastic premiums and returns, in which an insurer makes both risk-free and risky investments in two lines of business, and an accident may cause more than one claim. In this model, we allow that the two log-price processes are both real-valued Lévy processes, the claim numbers from the same business line, the two accident arrival processes and the two premium processes from two business lines are, respectively, arbitrarily dependent, and the premium processes are also arbitrarily dependent on all other random sources except the log-price processes. Under the condition that all claims from the same line are pairwise quasi-asymptotically independent and consistently varying-tailed, this paper establishes the asymptotic formulas for two types of finite-time ruin probabilities.
{"title":"Asymptotic behavior of finite-time ruin probabilities in a bidimensional compound risk model","authors":"Jinjin Zhang , Yang Yang , Lin Xu","doi":"10.1016/j.spl.2025.110529","DOIUrl":"10.1016/j.spl.2025.110529","url":null,"abstract":"<div><div>Consider a bidimensional compound risk model with stochastic premiums and returns, in which an insurer makes both risk-free and risky investments in two lines of business, and an accident may cause more than one claim. In this model, we allow that the two log-price processes are both real-valued Lévy processes, the claim numbers from the same business line, the two accident arrival processes and the two premium processes from two business lines are, respectively, arbitrarily dependent, and the premium processes are also arbitrarily dependent on all other random sources except the log-price processes. Under the condition that all claims from the same line are pairwise quasi-asymptotically independent and consistently varying-tailed, this paper establishes the asymptotic formulas for two types of finite-time ruin probabilities.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110529"},"PeriodicalIF":0.7,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864936","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}
Pub Date : 2025-08-18DOI: 10.1016/j.spl.2025.110533
José Manuel Corcuera, Rubén Jiménez
In this paper, we propose a novel generalisation of the signature of a path, motivated by fractional calculus, which is able to describe the solutions of linear Caputo controlled FDEs. We also propose another generalisation of the signature, inspired by the previous one, but more convenient to use in machine learning. Finally, we test this last signature in a toy application to the problem of handwritten digit recognition, where significant improvements in accuracy rates are observed compared to those of the original signature.
{"title":"Fractional signature: A generalisation of the signature inspired by fractional calculus","authors":"José Manuel Corcuera, Rubén Jiménez","doi":"10.1016/j.spl.2025.110533","DOIUrl":"10.1016/j.spl.2025.110533","url":null,"abstract":"<div><div>In this paper, we propose a novel generalisation of the signature of a path, motivated by fractional calculus, which is able to describe the solutions of linear Caputo controlled FDEs. We also propose another generalisation of the signature, inspired by the previous one, but more convenient to use in machine learning. Finally, we test this last signature in a toy application to the problem of handwritten digit recognition, where significant improvements in accuracy rates are observed compared to those of the original signature.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110533"},"PeriodicalIF":0.7,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864938","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}
Pub Date : 2025-08-18DOI: 10.1016/j.spl.2025.110528
Priti, Arun Kumar
In this article, we study the space–time fractional diffusion equation (STFDE) which is a generalization of the classical diffusion equation, in the presence of stochastic resetting. The STFDE is formulated by replacing the standard time and space derivatives with the Caputo and Riesz fractional derivatives, respectively, to capture anomalous diffusion behaviors. We derive analytical solutions using Laplace and Fourier transforms, and express them in terms of Fox H-functions. We obtain a closed-form expression for the stationary distribution and prove the finiteness of the mean first passage time. Additionally, we examine how stochastic resetting influences the infinite divisibility of the standard diffusion process, showing that this property is lost once resetting is introduced. The reset mechanism interrupts the Lévy process at random times, effectively altering the jump structure and destroying the self-decomposability required for infinite divisibility.
{"title":"Space–time fractional diffusion with stochastic resetting","authors":"Priti, Arun Kumar","doi":"10.1016/j.spl.2025.110528","DOIUrl":"10.1016/j.spl.2025.110528","url":null,"abstract":"<div><div>In this article, we study the space–time fractional diffusion equation (STFDE) which is a generalization of the classical diffusion equation, in the presence of stochastic resetting. The STFDE is formulated by replacing the standard time and space derivatives with the Caputo and Riesz fractional derivatives, respectively, to capture anomalous diffusion behaviors. We derive analytical solutions using Laplace and Fourier transforms, and express them in terms of Fox H-functions. We obtain a closed-form expression for the stationary distribution and prove the finiteness of the mean first passage time. Additionally, we examine how stochastic resetting influences the infinite divisibility of the standard diffusion process, showing that this property is lost once resetting is introduced. The reset mechanism interrupts the Lévy process at random times, effectively altering the jump structure and destroying the self-decomposability required for infinite divisibility.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110528"},"PeriodicalIF":0.7,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864935","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}
Pub Date : 2025-08-16DOI: 10.1016/j.spl.2025.110530
Bochen Jin
We investigate the limiting behaviour of the path of random bridges treated as random sets in with the Euclidean metric and the dimension increasing to infinity. The main result states that, in the square integrable case, the limit (in the Gromov–Hausdorff sense) is deterministic, namely, it is equipped with the pseudo-metric . We also show that, in the heavy-tailed case with summands regularly varying of order , the limiting metric space has a random metric derived from the bridge variant of a subordinator.
{"title":"Random bridges in spaces of growing dimension","authors":"Bochen Jin","doi":"10.1016/j.spl.2025.110530","DOIUrl":"10.1016/j.spl.2025.110530","url":null,"abstract":"<div><div>We investigate the limiting behaviour of the path of random bridges treated as random sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with the Euclidean metric and the dimension <span><math><mi>d</mi></math></span> increasing to infinity. The main result states that, in the square integrable case, the limit (in the Gromov–Hausdorff sense) is deterministic, namely, it is <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> equipped with the pseudo-metric <span><math><msqrt><mrow><mrow><mo>|</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>|</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mrow><mo>|</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></msqrt></math></span>. We also show that, in the heavy-tailed case with summands regularly varying of order <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, the limiting metric space has a random metric derived from the bridge variant of a subordinator.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110530"},"PeriodicalIF":0.7,"publicationDate":"2025-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864937","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}
Pub Date : 2025-08-14DOI: 10.1016/j.spl.2025.110524
Panxu Yuan , Zhenfeng Zou
Tsallis entropy is a commonly used uncertainty measure in information theory. In this paper, three new bounds of Tsallis entropy are provided for discrete random variables with finite support. This improves the understanding of the information content in complex systems.
{"title":"A note on three new bounds of Tsallis entropy","authors":"Panxu Yuan , Zhenfeng Zou","doi":"10.1016/j.spl.2025.110524","DOIUrl":"10.1016/j.spl.2025.110524","url":null,"abstract":"<div><div>Tsallis entropy is a commonly used uncertainty measure in information theory. In this paper, three new bounds of Tsallis entropy are provided for discrete random variables with finite support. This improves the understanding of the information content in complex systems.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110524"},"PeriodicalIF":0.7,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860661","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}
Pub Date : 2025-08-13DOI: 10.1016/j.spl.2025.110525
Dawei Lu, Yunpeng Shi, Junhan Zhao
Let be an -dependent sequence of events. In this paper, we proved the second part of the Borel–Cantelli lemma still holds under -dependence. Furthermore, we also obtained the quantitative version of this result.
{"title":"The Borel–Cantelli lemma under m-dependence","authors":"Dawei Lu, Yunpeng Shi, Junhan Zhao","doi":"10.1016/j.spl.2025.110525","DOIUrl":"10.1016/j.spl.2025.110525","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup></math></span> be an <span><math><mi>m</mi></math></span>-dependent sequence of events. In this paper, we proved the second part of the Borel–Cantelli lemma still holds under <span><math><mi>m</mi></math></span>-dependence. Furthermore, we also obtained the quantitative version of this result.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110525"},"PeriodicalIF":0.7,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860297","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}
Pub Date : 2025-08-13DOI: 10.1016/j.spl.2025.110523
Mario A. Estrada , Alex D. Ramos , Pablo M. Rodriguez
We study a stochastic process of multiple coalescing particles. In our process, particles begin at an arbitrary but fixed vertex of a complete graph. Each particle performs an independent discrete-time symmetric random walk on the graph. When two or more particles meet at a given vertex, they merge into a single particle that continues the random walk through the graph. If a particle jumps to a vertex that has been previously visited by another particle, it is removed from the system. We analyze the asymptotic behavior of the absorption time of the process; i.e., the number of steps until the last particle is removed from the system.
{"title":"On the mean absorption time of multiple coalescing particles with removal at previously visited vertices","authors":"Mario A. Estrada , Alex D. Ramos , Pablo M. Rodriguez","doi":"10.1016/j.spl.2025.110523","DOIUrl":"10.1016/j.spl.2025.110523","url":null,"abstract":"<div><div>We study a stochastic process of multiple coalescing particles. In our process, <span><math><mi>k</mi></math></span> particles begin at an arbitrary but fixed vertex of a complete graph. Each particle performs an independent discrete-time symmetric random walk on the graph. When two or more particles meet at a given vertex, they merge into a single particle that continues the random walk through the graph. If a particle jumps to a vertex that has been previously visited by another particle, it is removed from the system. We analyze the asymptotic behavior of the absorption time of the process; i.e., the number of steps until the last particle is removed from the system.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110523"},"PeriodicalIF":0.7,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860662","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}
Pub Date : 2025-08-12DOI: 10.1016/j.spl.2025.110519
Ruinan Li , Yumeng Li
We establish transportation cost-information inequalities for solutions of nonlinear stochastic partial differential equation of fractional order in both space and time variables with deterministic and bounded initial conditions: where , , , is the Caputo fractional derivative, is the fractional/power of Laplacian, is the Riemann–Liouville integral operator, is a space–time white noise, and is a bounded and Lipschitz function. Since the space variable is defined on the unbounded domain , the inequalities are proved under a weighted -norm in the spatial domain.
{"title":"Transportation cost-information inequality for non-linear time-fractional stochastic heat equation driven by space–time white noise","authors":"Ruinan Li , Yumeng Li","doi":"10.1016/j.spl.2025.110519","DOIUrl":"10.1016/j.spl.2025.110519","url":null,"abstract":"<div><div>We establish transportation cost-information inequalities <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>2</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> for solutions of nonlinear stochastic partial differential equation of fractional order in both space and time variables with deterministic and bounded initial conditions: <span><span><span><math><mrow><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msubsup><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mfenced><mrow><mi>σ</mi><mrow><mo>(</mo><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfenced><mspace></mspace><mspace></mspace><mtext>in</mtext><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></math></span>, <span><math><mrow><mi>γ</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, <span><math><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>β</mi></mrow></msubsup></math></span> is the Caputo fractional derivative, <span><math><mrow><mo>−</mo><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></math></span> is the fractional/power of Laplacian, <span><math><msubsup><mrow><mi>I</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup></math></span> is the Riemann–Liouville integral operator, <span><math><mrow><mover><mrow><mi>W</mi></mrow><mrow><mo>̇</mo></mrow></mover><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is a space–time white noise, and <span><math><mrow><mi>σ</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math></span> is a bounded and Lipschitz function. Since the space variable is defined on the unbounded domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, the inequalities are proved under a weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm in the spatial domain.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110519"},"PeriodicalIF":0.7,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860660","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}
Pub Date : 2025-08-10DOI: 10.1016/j.spl.2025.110522
Cyrille Joutard
Assuming real and independent and identically distributed observations, we obtain a classical pointwise saddlepoint approximation for the tail probability of the Parzen–Rosenblatt density estimator. This saddlepoint approximation is similar to the one which was first obtained by Daniels (1987) for the sample mean via the method of indirect Edgeworth expansion.
{"title":"Saddlepoint approximation for the kernel density estimator","authors":"Cyrille Joutard","doi":"10.1016/j.spl.2025.110522","DOIUrl":"10.1016/j.spl.2025.110522","url":null,"abstract":"<div><div>Assuming real and independent and identically distributed observations, we obtain a classical pointwise saddlepoint approximation for the tail probability of the Parzen–Rosenblatt density estimator. This saddlepoint approximation is similar to the one which was first obtained by Daniels (1987) for the sample mean via the method of indirect Edgeworth expansion.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"226 ","pages":"Article 110522"},"PeriodicalIF":0.7,"publicationDate":"2025-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828123","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}
Pub Date : 2025-08-08DOI: 10.1016/j.spl.2025.110521
Sadillo Sharipov
In this brief note, we study the strong law of large numbers for random walks in random scenery. Under the assumptions that the random scenery is non-stationary and satisfies weakly dependent condition with an appropriate rate, we establish strong law of large numbers for random walks in random scenery. Our results extend the known results in the literature.
{"title":"Strong law of large numbers for random walks in weakly dependent random scenery","authors":"Sadillo Sharipov","doi":"10.1016/j.spl.2025.110521","DOIUrl":"10.1016/j.spl.2025.110521","url":null,"abstract":"<div><div>In this brief note, we study the strong law of large numbers for random walks in random scenery. Under the assumptions that the random scenery is non-stationary and satisfies weakly dependent condition with an appropriate rate, we establish strong law of large numbers for random walks in random scenery. Our results extend the known results in the literature.</div></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"227 ","pages":"Article 110521"},"PeriodicalIF":0.7,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831478","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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