本文所探討的是關於稀分更新過程。首先給定一更新過程A，其到達間距為連續型的分佈，經過多項分佈的稀分後得到A_i , i =1,…, k，共k個稀分更新過程；我們分別對這些稀分更新過程的變異數、期望值、共變異數以及剩餘壽命作一些探討，並給出其與Poisson過程間的關係。另一方面，我們將推廣為考慮在每個到達時間允許有X個到達的情況，X為定義在正整數上之隨機變數。對於每一個到達，將會依隨機向量(β_1,…, β_k)的比例撕裂至k個新的過程，而得到D_i , i =1,…, k；其中(β_1,…, β_k)將會滿足P(β_1+…+ β_k=1)=1，且β_i有可能為負的。相同地，我們亦對這些新過程的變異數、期望值作一些探討，並探究其與Poisson過程間的關係。

In this thesis we investigate thinning of the renewal process. After multinomial thinning from a renewal process A, we obtain the k thinned processes, A_i , i =1,…, k. Based on some characterizations of the Poisson process as a renewal process, we give another characterizations of the Poisson process from some relations of expectation, variance, covariance, residual life of the k thinned processes. Secondly, we consider that at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the i-th new atom assigned to the process D_i , i =1,…, k. We also have characterizations of the Poisson process from some relations of expectation, variance of the process D_i , i =1,…, k.

<ol>
<li>Introduction</li>
<li>Preliminaries</li>
<li>Multinomial Thinning</li>
<li>Generalization of Multinomial Thinning</li>
</ol>

<ol>
<li>Chandramohan J, Foley RD, Disney RL (1985) Thinning of point processes-covariance analysis. <i>Adv. Appl. Prob.</i>&nbsp; <b>17</b>, 127-146.</li>
<li>Chandramohan J, Liang LK (1985) Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes. <i> J. Appl. Prob.</i> <b>22</b>, 828-835.</li>
<li>Chen YH (1994) Classes of life distributions and renewal counting processes. <i> J. Appl. Prob.</i> <b>23</b>, 233-235.</li>
<li>Grandell J (1976) <i> Doubly Stochastic Poisson Processes</i>. Lect. Notes Math. 529 Springer-Verlag, New York.</li>
<li>Gupta PL, Gupta RC (1986) A characterization of the Poisson process. <i>J. Appl. Prob.</i> <b>23</b>, 233-235.</li>
<li>Huang WJ (1989) On some characterizations of the Poisson process. <i> J. Appl. Prob.</i> <b>26</b>, 176-181.</li>
<li>Huang WJ, Li SH (1993) Characterizations of the Poisson process using the variance. <i> Commun. Statist.-Theory Meth.</i> <b>22</b>, 1371-1382.</li>
<li>Huang WJ, Li SH, Su JC (1993) Some characterizations of the Poisson process and geometric renewal process. <i> J. Appl. Prob.</i> <b>30</b>, 121-130.</li>
<li>Huang WJ, Su JC (1997) On a study of renewal process connected with certain conditional moments.<br><i> Sankhya Ser. A</i> <b>59</b>, 28-41.</li>
<li>Huang WJ, Chang WC (2000) On a study of the exponential and Poisson characteristics of the Poisson process. <i>Metrika</i> <b>50</b>, 247-254.</li>
<li>Karr AF (1986) <i>Point Processes and Their Statistical Inference.</i> Marcel Dekker, New York.</li>
<li>Li SH, Huang WJ, Huang MNL (1994) Characterizations of the Poisson process as a renewal process via two conditional moments. <i>Ann. Inst. Stat. Math.</i> <b>46</b>, 351-360.</li>
<li>Mecke J (1968) Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. <i>Z. Wahrschein. und Verw. Geb.</i> <b>11</b>, 74-81.</li>
<li>Thed&eacute;en T (1986) <i>The inverses of thinned point processes.</i> Dept. of Statistics, University of Stockholm, Research Report 1986:1.</li>
<li>Yannaros N (1988a) The Inverses of Thinned Renewal Processes. <i>J. Appl. Prob.</i> <b>25</b>, 822-828.</li>
<li>Yannaros N (1988b) On Cox Processes and Gamma Renewal Processes. <i>J. Appl. Prob.</i> <b>25</b>, 423-427.</li>
<li>Yannaros N (1989) On Cox and Renewal Processes. <i>Stat. Prob. Lett.</i> <b>7</b>, 431-433.</li>
</ol>