Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’offspring Xn with the mean 1+B(1+n)−β for β∈(0,1) and ‘displacement’ ξ...Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’offspring Xn with the mean 1+B(1+n)−β for β∈(0,1) and ‘displacement’ ξn with a drift A(1+n)^(−2α) for α∈(0,1/2), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ ξn is strictly positive or negative for |A|>0 but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter β and α.展开更多
In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(...In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ < m or λ > m, respectively;(2) when λ = m, 0 < α < 1,(Xn)n≥ 0 is transient/recurrent according to whether A < 0 or A > 0, respectively.In particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).展开更多
This paper proposes a kernel estimator for the coefficient of multidimensional time-varying diffusion processes as an extension of the estimation model for one dimensional diffusion coefficient to the multidimensional...This paper proposes a kernel estimator for the coefficient of multidimensional time-varying diffusion processes as an extension of the estimation model for one dimensional diffusion coefficient to the multidimensional case.By using"time division",the authors overcome the problem of sample observation in time varying model.In addition,the authors prove the strong consistency and limit distribution of the estimator.Finally,the authors test the performance of the estimator through a simulation experiment and an empirical application.展开更多
基金This work was supported by the National Key Research and Development Program of China(No.2020YFA0712900)the National Natural Science Foundation of China(Grant NO.11971062)the Fundamental Research Funds for the Central Universities Grant(No.N180503019).
文摘Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’offspring Xn with the mean 1+B(1+n)−β for β∈(0,1) and ‘displacement’ ξn with a drift A(1+n)^(−2α) for α∈(0,1/2), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ ξn is strictly positive or negative for |A|>0 but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter β and α.
基金supported by National Natural Science Foundation of China(Grant Nos.11531001 and 11626245)
文摘In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ < m or λ > m, respectively;(2) when λ = m, 0 < α < 1,(Xn)n≥ 0 is transient/recurrent according to whether A < 0 or A > 0, respectively.In particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).
基金supported by the National Natural Science Foundation of China under Grant Nos.11271189and 11201229。
文摘This paper proposes a kernel estimator for the coefficient of multidimensional time-varying diffusion processes as an extension of the estimation model for one dimensional diffusion coefficient to the multidimensional case.By using"time division",the authors overcome the problem of sample observation in time varying model.In addition,the authors prove the strong consistency and limit distribution of the estimator.Finally,the authors test the performance of the estimator through a simulation experiment and an empirical application.
