本论文分为三个部分. 第一部分, 我们主要研究非随机环境中的非紧邻随机游动. 这里要求游动往左的跳幅最大为~$L$, 往右的跳幅最大为~$R$~(简记为~$(L, R)$-模型). 在这一部分, 我们使用随机游动中的内蕴分枝结构这种新的工具得到了一些显式的结果.~(1)~对于半直线上状态非齐次的~$(L,1)$-模型, 通过计算游动在状态~$0$ 的占位时, 我们给出了过程常返暂留性的显式判定; 通过对游动的轨道进行分解, 我们计算出从任一状态~$i$ 出发, 首次返回~$i$ 所花费时间的期望, 进而给出了平稳分布的显式表达; 在游动正常返时, 我们研究了平稳分布的尾部渐进行为.~(2)~对于半直线上状态非齐次的~$(1,R)$-模型, 我们研究了当游动的转移概率带有某种渐进扰动时, 游动常返暂留性的变化, 即所谓的~Lamperti 问题.通过计算从~$0$ 出发首次返回~$0$ 时, 各状态访问次数的期望, 我们给出了正常返性的判别. 第二部分, 我们主要考虑随机环境中的非紧邻随机游动.~(1)~对于半直线上的~$(1,R)$-模型, 在构造出合适的李雅普诺夫函数后, 运用马氏链的鞅判别准则, 我们给出了过程常返暂留性以及正常返性的判定.~(2)~对于一类特殊的~$(\overline{L},1)$-模型~(即游动往左的跳幅只能为~$L$, 往右的跳幅只能为~$1$), 我们揭示了其中的内蕴拟分枝结构. 以此为工具, 我们运用~``从粒子看环境" 的方法证明了大数定律, 并给出了大数定律速度的显式表达. 此外, 我们还给出了半直线上该模型常返暂留性以及正常返性的判定. 第三部分, 我们研究有限状态马氏链首中时的分布问题. 对于一般有限状态的马氏链, 如果用~$\tau_{i,d}$ 表示从~$i$ 出发, 首次击中~$d$ 的时间, 我们运用轨道分解的办法给出了~$\tau_{i,d}$ 的母函数或拉普拉斯变换. 我们的结果覆盖了已有的紧邻或~skip-free 马氏链的相关结论. 此外, 我们还考虑了时间可逆的遍历马氏链从平稳分布出发时, 首次击中状态~$0$ 的分布.

This thesis consists of three parts. In the first part, we study the non-nearest random walk in non-random environments. We allow the random walk jumps to the left at a maximal size $L$, and jumps to the right at a maximal size $R$~(denoted by~$(L,R)$-model). Using the new tools~(the intrinsic branching structure within the random walk), we provide some explicit results.~(1)~For the $(L,1)$ state-dependent random walk on the half line, by calculating the expectation of the local time at zero, we obtain an explicit criterion for recurrence and transience. We calculate the expectation of the first return time to the starting point when the random walk starts at $i$ by decomposing the trajectory, as a consequence, we provide an explicit expression for the stationary distribution. Then, we study the asymptotic behavior of the stationary distribution.~(2)~For the $(1,R)$ state-dependent random walk on the half line, we study the Lamperti problem, the asymptotic behavior of the $(1,R)$-model with asymptotically zero drift. We give an explicit criterion for positive recurrence by calculating the expectation of the local time at each state before the first return to the starting point when the walk starts at $0$. In the second part, we study the non-nearest random walk in random environments~(RWRE for short).~(1)~For the $(1,R)$-RWRE on the half line, by constructing appropriate Lyapunov functions, we give an criterion for recurrence and transience by using the martingale criteria for countable irreducible Markov chains.~(2)~For a special $(\overline{L},1)$-RWRE, allowing jumps to the left with size $L$, and jumps to the right with size $1$. We reveal a intrinsic quasi-branching structure for this model. As an application, we prove the law of large numbers by a method known as ``the environment viewed from particles". Moreover, we give an explicit expression of the speed of the law of large numbers. In addition, we provide an criterion for recurrence and transience when the walk restricted on the half line. In the third part, we study the hitting time distribution for finite state Markov chains. Let $\tau_{i,d}$ be the absorbing time of state $d$ when the chain starts from $i$, by decomposing the trajectory, we give the generation functions or Laplace transforms of $\tau_{i,d}$. We recover the results for the nearest random walk and the skip-free Markov chains. After that, we consider the hitting time distribution of a time reversible ergodic Markov chain which starts from the stationary distribution.