This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts servi...This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.展开更多
Spinor Bose–Einstein condensates(BECs)are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom.This study concentrates on a(1+1)-dimensional t...Spinor Bose–Einstein condensates(BECs)are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom.This study concentrates on a(1+1)-dimensional three-couple Gross–Pitaevskii system to depict the macroscopic spinor BEC waves within the meanfield approximation.Regarding the distribution of the atoms corresponding to the three vertical spin projections,a known binary Darboux transformation is utilized to derive the𝑁matter-wave soliton solutions and triple-pole matter-wave soliton solutions on the zero background,where𝑁is a positive integer.For those multiple matterwave solitons,the asymptotic analysis is performed to obtain the algebraic expressions of the soliton components in the𝑁matter-wave solitons and triple-pole matter-wave solitons.The asymptotic results indicate that the matter-wave solitons in the spinor BECs possess the property of maintaining their energy content and coherence during the propagation and interactions.Particularly,in the𝑁matter-wave solitons,each soliton component contributes to the phase shifts of the other soliton components;and in the triple-pole matter-wave solitons,stable attractive forces exist between the different matter-wave soliton components.Those multiple matter-wave solitons are graphically illustrated through three-dimensional plots,density plot and contour plot,which are consistent with the asymptotic analysis results.The present analysis may provide the explanations for the complex natural mechanisms of the matter waves in the spinor BECs,and may have potential applications in designs of atom lasers,atom interferometry and coherent atom transport.展开更多
Let {X_i;i≥1} be a strictly stationary sequence of associated random variables with mean zero and let σ2=EX2_1+2∞_~j=2 EX_1X_j with 0<σ2<∞.Set S_n=n_~i=1 X_i,the precise asymptotics for _~n≥1 n^rp-2 P(|S_n...Let {X_i;i≥1} be a strictly stationary sequence of associated random variables with mean zero and let σ2=EX2_1+2∞_~j=2 EX_1X_j with 0<σ2<∞.Set S_n=n_~i=1 X_i,the precise asymptotics for _~n≥1 n^rp-2 P(|S_n|≥εn^1p ),_~n≥1 1nP(|S_n|≥εn^1p ) and _~n≥1 (log n)δnP(|S_n|≥εnlogn) as ε0 are established.展开更多
Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers s...Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.展开更多
In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑...In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.展开更多
This article gives the equivalent conditions of the local asymptotics for the overshoot of a random walk with heavy-tailed increments, from which we find that the above asymptotics are different from the local asympto...This article gives the equivalent conditions of the local asymptotics for the overshoot of a random walk with heavy-tailed increments, from which we find that the above asymptotics are different from the local asymptotics for the supremum of the random walk. To do this, the article first extends and improves some existing results about the solutions of renewal equations.展开更多
In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions a...In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.展开更多
Let Ω be a non-empty bounded open set in Rn(n ≥1) with boundary (?)Ω=Γ1∪Γ2. WedefineIn this paper, we consider the following variational eigenvalue problem:where △ denotes the Laplacian in Ω. We say that the s...Let Ω be a non-empty bounded open set in Rn(n ≥1) with boundary (?)Ω=Γ1∪Γ2. WedefineIn this paper, we consider the following variational eigenvalue problem:where △ denotes the Laplacian in Ω. We say that the scalar λ is an eigenvalue of (P) if展开更多
Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w...Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w); |w|>1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, L n(·) means the leading coefficient of the orthonormal matrix polynomials Φ n(z;·). Finally, we deduce the asymptotic behavior of Φ n(w;)Φ n(w;Ω)* in the case when M=I.展开更多
In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomial...In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Delft ＆ Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior.展开更多
In this article, we use penalized spline to estimate the hazard function from a set of censored failure time data. A new approach to estimate the amount of smoothing is provided. Under regularity conditions we establi...In this article, we use penalized spline to estimate the hazard function from a set of censored failure time data. A new approach to estimate the amount of smoothing is provided. Under regularity conditions we establish the consistency and the asymptotic normality of the penalized likelihood estimators. Numerical studies and an example are conducted to evaluate the performances of the new procedure.展开更多
In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearing loss injury among subjects in a crowd with a wide spectrum of heterogeneous individual injury susceptibility due to biovariability. The injury r...In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearing loss injury among subjects in a crowd with a wide spectrum of heterogeneous individual injury susceptibility due to biovariability. The injury risk of a crowd is defined as the average fraction of injured. We examined mathematically the injury risk of a crowd vs the number of acoustic impulses the crowd is exposed to, under the assumption that all impulses act independently in causing injury regardless of whether one is preceded by another. We concluded that the observed dose-response relation can be explained solely on the basis of biovariability in the form of heterogeneous susceptibility. We derived an analytical solution for the distribution density of injury susceptibility, as a power series expansion in terms of scaled log individual non-injury probability. While theoretically the power series converges for all argument values, in practical computations with IEEE double precision, at large argument values, the numerical accuracy of the power series summation is completely wiped out by the accumulation of round-off errors. In this study, we derive a general asymptotic approximation at large argument values, for the distribution density. The combination of the power series and the asymptotics provides a practical numerical tool for computing the distribution density. We then use this tool to verify numerically that the distribution obtained in our previous theoretical study is indeed a proper density. In addition, we will also develop a very efficient and accurate Pade approximation for the distribution density.展开更多
A general weighted second order elliptic equation involving critical growth is considered on bounded smooth. domain in n-dimension space. There is the singular point for the weighted coefficients in the domain. With g...A general weighted second order elliptic equation involving critical growth is considered on bounded smooth. domain in n-dimension space. There is the singular point for the weighted coefficients in the domain. With generalized blow up method, some results are obtained for asymptotic behavior of positive solutions. This problem includes Laplacian operators as special cases.展开更多
Let Y_i=M(X_i)+ei, where M(x)=E(Y|X=x) is an unknown realfunction on B(? R), {(X_1,Y_i)} is a stationary and m(n)-dependent sample from(X, Y), the residuals {e_i} are independent of {X_i} and have unknown common densi...Let Y_i=M(X_i)+ei, where M(x)=E(Y|X=x) is an unknown realfunction on B(? R), {(X_1,Y_i)} is a stationary and m(n)-dependent sample from(X, Y), the residuals {e_i} are independent of {X_i} and have unknown common densityf(x). In [2] a nonparametric estimate f_n(x) for f(x) has been proposed on the basisof the residuals estimates. In this paper, we further obtain the asymptotic normalityand the law of the iterated logarithm of f_n(x) under some suitable conditions. Theseresults together with those in [2] bring the asymptotic theory for the residuals densityestimate in nonparametric regression under m(n)-dependent sample to completion.展开更多
This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.
In this paper, we give precise formulas for the general two-dimensional recursion sequences by generating function method, and make use of the multivariate generating functions asymptotic estimation technique to compu...In this paper, we give precise formulas for the general two-dimensional recursion sequences by generating function method, and make use of the multivariate generating functions asymptotic estimation technique to compute their asymptotic values.展开更多
The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the conver...The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the convergence for the module of minimizers are presented.展开更多
In this paper, we discuss the precise asymptotics of moving-average process Xt =∞∑j=0 ajEt-j under some suitable conditions, where {εt, t∈ Z} is a sequence j=0 of stationary ALNQD random variables with mean zeros...In this paper, we discuss the precise asymptotics of moving-average process Xt =∞∑j=0 ajEt-j under some suitable conditions, where {εt, t∈ Z} is a sequence j=0 of stationary ALNQD random variables with mean zeros and finite variances.展开更多
In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the ba...In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.展开更多
This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectiv...This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectively,are real-valued,dependent and heavy-tailed,while the random weights{Θi,θi;i∈N}are nonnegative and arbitrarily dependent,but the three sequences{X_(i);i∈N},{Y_(i);i∈N}and{Θ_(i),θ_(i);i∈N}are mutually independent.Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights,we establish some(uniformly)asymptotic formulas for the joint tail probability of the two randomly weighted sums,expressing the insensitivity with respect to the underlying weak dependence structures.As applications,we consider both discrete-time and continuous-time insurance risk models,and obtain some asymptotic results for ruin probabilities.展开更多
文摘This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.
基金work was supported by the National Natural Science Foundation of China(Grant No.12161061)the Fundamental Research Funds for the Inner Mongolia University of Finance and Economics(Grant No.NCYWT23036)+2 种基金the Young innovative and Entrepreneurial Talents of the Inner Mongolia Grassland Talents Project in 2022,Autonomous Region“Five Ma-jor Tasks"Research Special Project for the Inner Mongo-lia University of Finance and Economics in 2024(Grant No.NCXWD2422)High Quality Research Achievement Cultivation Fund for the Inner Mongolia University of Fi-nance and Economics in 2024(Grant No.GZCG2426)the Talent Development Fund of Inner Mongolia.
文摘Spinor Bose–Einstein condensates(BECs)are formed when atoms in the multi-component BECs possess single hyperfine spin states but retain internal spin degrees of freedom.This study concentrates on a(1+1)-dimensional three-couple Gross–Pitaevskii system to depict the macroscopic spinor BEC waves within the meanfield approximation.Regarding the distribution of the atoms corresponding to the three vertical spin projections,a known binary Darboux transformation is utilized to derive the𝑁matter-wave soliton solutions and triple-pole matter-wave soliton solutions on the zero background,where𝑁is a positive integer.For those multiple matterwave solitons,the asymptotic analysis is performed to obtain the algebraic expressions of the soliton components in the𝑁matter-wave solitons and triple-pole matter-wave solitons.The asymptotic results indicate that the matter-wave solitons in the spinor BECs possess the property of maintaining their energy content and coherence during the propagation and interactions.Particularly,in the𝑁matter-wave solitons,each soliton component contributes to the phase shifts of the other soliton components;and in the triple-pole matter-wave solitons,stable attractive forces exist between the different matter-wave soliton components.Those multiple matter-wave solitons are graphically illustrated through three-dimensional plots,density plot and contour plot,which are consistent with the asymptotic analysis results.The present analysis may provide the explanations for the complex natural mechanisms of the matter waves in the spinor BECs,and may have potential applications in designs of atom lasers,atom interferometry and coherent atom transport.
文摘Let {X_i;i≥1} be a strictly stationary sequence of associated random variables with mean zero and let σ2=EX2_1+2∞_~j=2 EX_1X_j with 0<σ2<∞.Set S_n=n_~i=1 X_i,the precise asymptotics for _~n≥1 n^rp-2 P(|S_n|≥εn^1p ),_~n≥1 1nP(|S_n|≥εn^1p ) and _~n≥1 (log n)δnP(|S_n|≥εnlogn) as ε0 are established.
基金National Natural Science Foundation of China(10571073).
文摘Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.
文摘In the case of Z＋^d（d ≥ 2）-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k∈ Z＋^d} i.i.d, random variables with mean 0, Sn =∑k≤nXk and Vn^2 = ∑j≤nXj^2, the precise asymptotics for ∑n1/｜n｜（log｜n｜dP（｜Sn/Vn｜≥ε√log log｜n｜） and ∑n（logn｜）b/｜n｜（log｜n｜）^d-1P（｜Sn/Vn｜≥ε√log n）,as ε↓0,is established.
基金supported by National Science Foundation of China(10671139, 11071182, 10771119)Natural Science Foundation of Jiangsu Highter Eduction Institution of China(10KJB110010)the research foundation of SUST
文摘This article gives the equivalent conditions of the local asymptotics for the overshoot of a random walk with heavy-tailed increments, from which we find that the above asymptotics are different from the local asymptotics for the supremum of the random walk. To do this, the article first extends and improves some existing results about the solutions of renewal equations.
文摘In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.
基金The NNSF (10025107) of China and the 973 Projects.
文摘Let Ω be a non-empty bounded open set in Rn(n ≥1) with boundary (?)Ω=Γ1∪Γ2. WedefineIn this paper, we consider the following variational eigenvalue problem:where △ denotes the Laplacian in Ω. We say that the scalar λ is an eigenvalue of (P) if
文摘Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L n()L n(Ω) -1 and Φ n(z;)Φ n(z;Ω) -1 where(z)=Ω(z)+Mδ(z-w); |w|>1,M is a positive definite matrix and δ is the Dirac matrix measure. Here, L n(·) means the leading coefficient of the orthonormal matrix polynomials Φ n(z;·). Finally, we deduce the asymptotic behavior of Φ n(w;)Φ n(w;Ω)* in the case when M=I.
基金supported in part by the National Natural Science Foundation of China (10471154 and 10871212)
文摘In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Delft ＆ Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior.
基金supported by the Natural Science Foundation of China(10771017,10971015,10231030)Key Project to Ministry of Education of the People’s Republic of China(309007)
文摘In this article, we use penalized spline to estimate the hazard function from a set of censored failure time data. A new approach to estimate the amount of smoothing is provided. Under regularity conditions we establish the consistency and the asymptotic normality of the penalized likelihood estimators. Numerical studies and an example are conducted to evaluate the performances of the new procedure.
文摘In our recent work (Wang, Burgei, and Zhou, 2018) we studied the hearing loss injury among subjects in a crowd with a wide spectrum of heterogeneous individual injury susceptibility due to biovariability. The injury risk of a crowd is defined as the average fraction of injured. We examined mathematically the injury risk of a crowd vs the number of acoustic impulses the crowd is exposed to, under the assumption that all impulses act independently in causing injury regardless of whether one is preceded by another. We concluded that the observed dose-response relation can be explained solely on the basis of biovariability in the form of heterogeneous susceptibility. We derived an analytical solution for the distribution density of injury susceptibility, as a power series expansion in terms of scaled log individual non-injury probability. While theoretically the power series converges for all argument values, in practical computations with IEEE double precision, at large argument values, the numerical accuracy of the power series summation is completely wiped out by the accumulation of round-off errors. In this study, we derive a general asymptotic approximation at large argument values, for the distribution density. The combination of the power series and the asymptotics provides a practical numerical tool for computing the distribution density. We then use this tool to verify numerically that the distribution obtained in our previous theoretical study is indeed a proper density. In addition, we will also develop a very efficient and accurate Pade approximation for the distribution density.
文摘A general weighted second order elliptic equation involving critical growth is considered on bounded smooth. domain in n-dimension space. There is the singular point for the weighted coefficients in the domain. With generalized blow up method, some results are obtained for asymptotic behavior of positive solutions. This problem includes Laplacian operators as special cases.
基金Project Supported by National Natural Science Foundation of China.
文摘Let Y_i=M(X_i)+ei, where M(x)=E(Y|X=x) is an unknown realfunction on B(? R), {(X_1,Y_i)} is a stationary and m(n)-dependent sample from(X, Y), the residuals {e_i} are independent of {X_i} and have unknown common densityf(x). In [2] a nonparametric estimate f_n(x) for f(x) has been proposed on the basisof the residuals estimates. In this paper, we further obtain the asymptotic normalityand the law of the iterated logarithm of f_n(x) under some suitable conditions. Theseresults together with those in [2] bring the asymptotic theory for the residuals densityestimate in nonparametric regression under m(n)-dependent sample to completion.
文摘This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.
文摘In this paper, we give precise formulas for the general two-dimensional recursion sequences by generating function method, and make use of the multivariate generating functions asymptotic estimation technique to compute their asymptotic values.
文摘The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the convergence for the module of minimizers are presented.
文摘In this paper, we discuss the precise asymptotics of moving-average process Xt =∞∑j=0 ajEt-j under some suitable conditions, where {εt, t∈ Z} is a sequence j=0 of stationary ALNQD random variables with mean zeros and finite variances.
基金supported by the National Natural Science Foundation of China(No.12175069 and No.12235007)Science and Technology Commission of Shanghai Municipality(No.21JC1402500 and No.22DZ2229014)Natural Science Foundation of Shanghai(No.23ZR1418100)。
文摘In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.
基金supported by the Humanities and Social Sciences Foundation of the Ministry of Education of China(Grant No.20YJA910006)Natural Science Foundation of Jiangsu Province of China(Grant No.BK20201396)+2 种基金supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province of China(Grant No.KYCX211939)supported by the Research Grants Council of Hong KongChina(Grant No.HKU17329216)。
文摘This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectively,are real-valued,dependent and heavy-tailed,while the random weights{Θi,θi;i∈N}are nonnegative and arbitrarily dependent,but the three sequences{X_(i);i∈N},{Y_(i);i∈N}and{Θ_(i),θ_(i);i∈N}are mutually independent.Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights,we establish some(uniformly)asymptotic formulas for the joint tail probability of the two randomly weighted sums,expressing the insensitivity with respect to the underlying weak dependence structures.As applications,we consider both discrete-time and continuous-time insurance risk models,and obtain some asymptotic results for ruin probabilities.
