# Some Limit Theorems in Geometric Processes

Some Limit Theorems in Geometric Processes Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {X n , n = 1, 2, · · ·} for which there exists a real number a > 0, such that {a n−1 X n , n = 1, 2, · · ·} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S n with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Some Limit Theorems in Geometric Processes

, Volume 19 (3) – Mar 3, 2017

## Some Limit Theorems in Geometric Processes

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 405–416 1 2 3 Yeh Lam ,Yao-hui Zheng , Yuan-lin Zhang Northeastern University at Qinhuangdao, 066004, China & Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong (E-mail: ylam@hkustasc.hku.hk) Department of Mathematics, Xiamen University, Xiamen 361005, China Institute of Applied Probability, Sanjiang University, Nanjing 210018, China & Department of Applied Math- ematics, Southeast University, Nanjing 210018, China [4,5] Abstract Geometric process (GP) was introduced by Lam , it is deﬁned as a stochastic process {X ,n = n−1 1, 2, ···} for which there exists a real number a> 0, such that {a X ,n =1, 2, ···} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We ﬁrst derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S with a> 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t. Keywords Geometric process, new better than used in expectation, stochastic order 2000 MR Subject Classiﬁcation 60G55, 60K99 1 Introduction In maintenance problems, it is often to assume that a failed system after repair will be “as good as new”, this is the perfect repair model. In practice, it is not always the case. A more reasonable model is the minimal repair model which assumes that a failed system after repair will function again, but with the same failure rate and the same eﬀective age as at the time of failure (see e.g. [1] or [2] for further details). An alternative approach is to apply a monotone process model. In practice, most systems are deteriorating due to ageing eﬀect and accumulated wearing. As a result, the successive operating...

/lp/springer_journal/some-limit-theorems-in-geometric-processes-jGFbAGll2T
Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0115-1
Publisher site
See Article on Publisher Site

### Abstract

Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {X n , n = 1, 2, · · ·} for which there exists a real number a > 0, such that {a n−1 X n , n = 1, 2, · · ·} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S n with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

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