本論文討論關於更新過程的相關問題。更仔細地說，令$gamma_{t}$代表一更新過程
A={A(t),t>0}的剩餘壽命。若$Var(gamma_{t})=E^2(gamma_{t})-E(gamma_{t})$，
則當到達間距為離散時，此更新過程為幾何更新過程。另一方面，藉由更新過程隨機和的尾部之討論，
證明隨機和的k次方仍滿足舊的比新的好之性質。

In this thesis we present some related problems about the renewal processes. More precisely, let $gamma_{t}$ be the residual life at time $t$ of the renewal process $A={A(t),t geq 0}$, $F$ be the common distribution function of the inter-arrival times. Under suitable conditions, we prove that if $Var(gamma_{t})=E^2(gamma_{t})-E(gamma_{t}),forall t=0,1 ho,2 ho,3 ho,... $, then $F$ will be geometrically distributed under the assumption $F$ is discrete. We also discuss
the tails of random sums for the renewal process. We prove that the $k$ power of random sum is always new worse than used ($NWU$).

1. Introduction
2. Preliminary
3. Characterization related to the geometric characteristic
4. A class of random sums and its NWU property
5. Discussion

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