变化环境下Galton-Watson过程的极限理论

研究了变化环境下的Galton-Watson过程.当环境的期望和方差有限时,给出了几乎处处环境下的首达时极限定理,以及log Z_n的大偏差定理.     

独立同分布随机环境中两性Galton-Watson分支过程灭绝概率的渐近行为

本文研究了独立同分布随机环境中的两性Galton-Watson分支过程,在上临界情形下,当k充分大时,qk≤ck<'-α>.     

Galton-Watson分支过程的谱半径的概率刻画

谱半径是不可约马尔可夫链的一个很重要的特征数字.Galton-Watson分支过程是一类特殊的马尔可夫链,我们已经证明了在不可约的条件下,Galton-Watson分支过程的谱半径等于其对应的概率母函数f(s)在灭绝概率q点的导数值.本文主要从理论上刻画从过程的任何状态逃离速度的Galton-Watson分支过程的谱半径的概率意义.     

反射扩散过程的某些极限定理

设（Ｘｉ）是由如下随机微分方程所决定的反射扩散过程： Ｘｔ＝Ｘ０＋∫＾ｔ０σ（Ｘｓ）ｄＷｓ＋∫＾ｔ０ｂ（Ｘｓ）ｄｓ＋Ｌｔ－Ｕｔ， Ｌｔ＝∫＾ｔ０Ｉ｛０｝（Ｘｓ）ｄＬｓ，Ｕｔ＝∫＾ｔ０Ｉ｛１｝（Ｘｓ）ｄＵｓ。 本文证明了当ｔ→∞时，Ｐｘ｛Ｘｔ∈Ａ｝→π（Ａ），１／ｔＥｘ（Ｌｒ）→ａ，１／ｔ     

两指标Levy过程的强极限定理

本文研究了两指标Ｌｅｖ过程增量平方和的强极限，获得了一些重要结果。     

一类暂留稳定过程的极限定理

本文研究了Ａ型暂留稳定过程在无穷远处的收敛速度，给出了一个重对数律．同时我们也得出了这类过程在起始点附近的一些性质．这些性质推广了［１］中的结果     

树上路径过程的随机路径条件概率的强极限定理

本文研究了树上路径过程随机转移概率和状态序偶出现频率的强极限定理.通过利用若干重要不等式,获得了树上路径过程的随机路径条件概率用不等式表示的几何平均强极限定理以及树上路径过程关于状态序偶出现频率的用不等式表示的强极限定理,所得结果推广了树上马氏链及非齐次马氏链中的结果.     

改进两参数Wiener过程若干极限定理

本文将讨论一类形为Ｗ（ｘ，ｎ）的两参数Ｗｉｅｎｅｒ过程．对于这类Ｗｉｅｎｅｒ过程的增量我们将找出适当的正则化因子β_ｎ和μ_ｎ，使得（ｎ）的极限为１．并且求出下列各极限及     

半鞅序列积分误差的极限过程的收敛定理

Jacod, Jakubowski和M\'emin讨论了与单个独立增量过程$X$的误差过程$^n\!X =X_t-X_{[nt]/n}$相关的积分误差过程$Y^n(X)$和$Z^{n,p}(X)$, 研究了半鞅序列$\{(nY^n(X),nZ^{n,p}(X))\}_{n\ge 1}$的极限定理. 记半鞅序列$\{(nY^n(X),nZ^{n,p}(X))\}_{n\ge1}$的极限过程为$(Y(X),Z^p(X))$, Jacod等给出了其极限过程$(Y(X)$, $Z^p(X))$的表达式. 本文将研究半鞅序列$\{X^n\}_{n\ge1}$积分误差的极限过程$Y(X^n)$和$Z^{p}(X^n)$的收敛定理, 主要研究半鞅序列$\{(X^n,Y(X^n),Z^p(X^n))\}_{n\ge1}$的依分布弱收敛和依分布稳定收敛.     

Limit Theorems for Multiplicative Processes

Let W be a non-negative random variable with EW=1, and let {W _i } be a family of independent copies of W, indexed by all the finite sequences i=i ₁i _n of positive integers. For fixed r and n the random multiplicative measure ⁿ _r has, on each r-adic interval at nth level, the density with respect to the Lebesgue measure on [0,1]. If EW log Wr, the sequence { ⁿ _r } _n converges a.s. weakly to the Mandelbrot measure _r . For each fixed 1n, we study asymptotic properties for the sequence of random measures { ⁿ _r } _r as r. We prove uniform laws of large numbers, functional central limit theorems, a functional law of iterated logarithm, and large deviation principles. The function-indexed processes is a natural extension to a tree-indexed process at nth level of the usual smoothed partial-sum process corresponding to n=1. The results extend the classical ones for { ¹ _r } _r , and the recent ones for the masses of { _r } _r established in Ref. 23.     

Limit Theorems for Stopped Functionals of Markov Renewal Processes

Some results for stopped random walks are extended to the Markov renewal setup where the random walk is driven by a Harris recurrent Markov chain. Some interesting applications are given; for example, a generalization of the alternating renewal process.     

A Regularity Condition and Strong Limit Theorems for Linear Birth–Growth Processes

A linear birth–growth process is generated by an inhomogeneous Poisson process on ℝ × [0, ∞). Seeds are born randomly according to the Poisson process. Once a seed is born, it commences immediately to grow bidirectionally with a constant speed. The positions occupied by growing intervals are regarded as covered. New seeds continue to form on the uncovered part of ℝ. This paper shows that the total number of seeds born on a very long interval satisfies the strong invariance principle and some other strong limit theorems. Also, a gap (an unproved regularity condition) in the proof of the central limit theory in [5] is filled in.     

Strong Law of Large Numbers and Central Limit Theorems for Functionals of Inhomogeneous Semi-Markov Processes

Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of Pyke Schaufele (Limit theorems for Markov renewal processes, Ann. Math. Statist., 35(4):1746–1764, 1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example, in finance and insurance. Unfortunately, no limit theorems have been obtained for functionals of inhomogeneous Markov renewal and semi-Markov processes as of today, to the best of the authors’ knowledge. In this article, we provide strong law of large numbers and central limit theorem results for such processes. In particular, we make an important connection of our results with the theory of ergodicity of inhomogeneous Markov chains. Finally, we provide an application to risk processes used in insurance by considering a inhomogeneous semi-Markov version of the well-known continuous-time Markov chain model, widely used in the literature.     

马尔可夫骨架过程的极限分布

本文运用基本更新定理和Smith关键更新定理等理论和方法,对马尔可夫骨架过程的极限分布进行深入研究,得到主要结果如下:去掉了原有结果中要求的绝对连续的条件,给出了马尔可夫骨架过程极限分布存在的充分条件;得到了马尔可夫骨架过程极限分布的具体公式,并证明了该极限分布为概率分布.     

Limit Theorems for the Non-linear Functional of Stationary Gaussian Processes

In this paper we consider two functional limit theorems for the non-linear functional of the stationary Gaussian process satisfying short range dependence conditions: the functional CLT for partial sum processes and the uniform CLT for a special class of functions. To carry out the proofs, we develop Rosenthal type inequalities for the functional of Gaussian processes.     

随机截断数据下乘积限过程的振动模

文中研究了随机截断数据下的剩积限过程的振动行为，证明了其振动模的收敛速度与完全样本下经验过程振动模的收敛速度相一致。