On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables
On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables
复制标题
离散随机变量序列模式等待时间问题的生成函数
DOI：
10.1023/a:1003756712643
复制
发表时间：
1998
影响因子：
1
通讯作者：
Masayuki Uchida
中科院分区：
文献类型：
--
作者：
Masayuki Uchida
AbstractLet X
$$_1$$
选项卡页面上创建
$$_2$$
, ...是独立同分布的随机变量序列，其取值于可数集S = {0，1，2，.}。我们所说的模式是指S中的有限元素序列。对于每个i = 0，1，2，...，我们用P表示
$$_i$$
=“a”
$$_{1}$$
一
$$_{i，2}$$
...一
$$_{i，k_i }$$
“长度为k的模式
$$_i$$
，E
$$_i$$
表示图案P
$$_i$$
发生在序列X中
$$_1$$
选项卡页面上创建
$$_2$$
, ....本文导出了直到第r次事件发生的等待时间分布的广义概率母函数
$$\{ E_i \} _{i = 0}^\infty$$
.我们还导出了高阶马氏链中长为l（l < k）的子模式出现次数分布的概率母函数，直到长为k的子模式出现为止。
AbstractLet X
$$_1$$
, X
$$_2$$
, ... be a sequence of independent and identically distributed random variables, which take values in a countable set S = {0, 1, 2, ...}. By a pattern we mean a finite sequence of elements in S. For every i = 0, 1, 2, ..., we denote by P
$$_i$$
= "a
$$_{i,1}$$
a
$$_{i,2}$$
... a
$$_{i,k_i }$$
" the pattern of some length k
$$_i$$
, and E
$$_i$$
denotes the event that the pattern P
$$_i$$
occurs in the sequence X
$$_1$$
, X
$$_2$$
, .... In this paper, we have derived the generalized probability generating functions of the distributions of the waiting times until the r-th occurrence among the events
$$\{ E_i \} _{i = 0}^\infty$$
. We also have derived the probability generating functions of the distributions of the number of occurrences of sub-patterns of length l(l < k) until the fiurrence of the pattern of length k in the higher order Markov chain.