Further limit theorems for the range of random walk
Further limit theorems for the range of random walk
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DOI：
10.1007/bf02788644
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发表时间：
1974-12
期刊：
影响因子：
--
通讯作者：
N. Jain;W. Pruitt
中科院分区：
文献类型：
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作者：
N. Jain;W. Pruitt
令 {X,, n> 1} 为独立同分布随机变量序列，定义在概率空间 (~,~', P) 上，其取 d 维整数格 E d 中的值。由S o= 0, Sn=~'~= 1 Xk 定义的序列{S,,n>0}称为随机游走。随机游走可以发生在 En 的真子群上。在这种情况下，子群同构于某个 Ek，k< d；如果 k< d，则应进行变换并在 k 维中考虑问题。如有必要，我们将在整篇论文中假设已经进行了这种简化，并且 d 是随机游走的真实维度。设p=P{St~0,S2~0,...}。如果 p> 0，随机游走称为瞬态，否则称为循环随机游走。瞬态的一个等价标准是级数~]~t P [S.= 0]的收敛。若E~=1~]k~。 P[Sk=0]收敛，则随机游走称为强瞬态。如果 d> 3，则所有随机游走都是瞬态的；如果 d> 5，则所有随机游走都是瞬态的，但瞬态和强瞬态随机游走也发生在较低维度中。随机游走 {S.} 到时间 n 的范围（用 R. 表示）只是集合 {So, S~,"', Sn} 的基数。如果 p= 1，则 R.= n+ 1 as 并且我们知道有关 Rn 的所有信息。对于 p< 1 的强瞬态随机游走[3] 中表明，对于某些 a2> 0，Var R.（R. 的方差）渐近 a2n，并且 (R.-ER.)/an 89 呈渐近正态分布，均值为 0，方差为 1。如果 EXx= O, E [Xx 12< co,
Let {X,, n> 1} be a sequence of independent identically distributed random variables, defined on a probability space (~,~', P), which take values in the d-dimensional integer lattice E d. The sequence {S,, n> 0} defined by S o= 0, Sn=~'~= 1 Xk is called a random walk. The random walk may take place on a proper subgroup of E n. In this case, the subgroup is isomorphic to some Ek, k< d; if k< d, then a transformation should be made and the problem considered in k dimensions. We will assume throughout the paper that this reduction has been made, if necessary, and that d is the genuine dimension of the random walk. Let p= P {S t~ 0, S 2~ 0,...}. The random walk is called transient if p> 0 and recurrent otherwise. An equivalent criterion for transience is the convergence of the series~]~ t P [S.= 0]. If E~= 1~] k~. P [Sk= 0] converges, then the random walk is called strongly transient. All random walks are transient if d> 3 and strongly transient if d> 5, but transient and strongly transient random walks also occur in lower dimensions.The range of the random walk {S.} up to time n, denoted by R., is simply the cardinality of the set {So, S~,"', Sn}. If p= 1, then R.= n+ 1 as and we know everything there is to know about Rn. For a strongly transient random walk with p< 1 it was shown in [3] that Var R.(variance of R.) is asymptotically a2n for some a2> 0 and (R.-ER.)/an 89 is asymptotically normally distributed with mean 0 and variance 1. If EXx= O, E [Xx 12< co,