In this paper, we study the count of head runs up to a fixed time in a two-state stationary Markov chain. We prove that in total variance distance, the negative binomial, Poisson and binomial distributions are appropr...In this paper, we study the count of head runs up to a fixed time in a two-state stationary Markov chain. We prove that in total variance distance, the negative binomial, Poisson and binomial distributions are appropriate approximations according to the relation of the variance and mean of the count, generalizing earlier results in previous literatures. The proof is based on Stein's method and coupling.展开更多
This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient con...This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace （relation） below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspaee, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11071021)
文摘In this paper, we study the count of head runs up to a fixed time in a two-state stationary Markov chain. We prove that in total variance distance, the negative binomial, Poisson and binomial distributions are appropriate approximations according to the relation of the variance and mean of the count, generalizing earlier results in previous literatures. The proof is based on Stein's method and coupling.
基金supported by National Natural Science Foundation of China(Grant No.11571202)the China Scholarship Council(Grant No.201406220019)
文摘This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace （relation） below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspaee, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.
