Sains Malaysiana 49(4)(2020): 919-928

http://dx.doi.org/10.17576/jsm-2020-4904-21

Limit Theorem for A Semi-Markovian Random Walk with General Interference of Chance

(Had Teorem untuk Jalan Rawak Semi-Markovan dengan Kemungkinan Gangguan Umum)

TAHIR KHANIYEV^1,3 & OZLEM ARDIC SEVINC^1,2*

¹Department of Industrial Engineering, TOBB University of Economics and Technology, 06560, Ankara, Turkey

²Department of Structural Economic Research, Central Bank of the Republic of Turkey, 06050, Altindag, Ankara, Turkey

³Institute of Control Systems, Azerbaijan National Academy of Sciences, AZ 1141, Baku, Azerbaijan

Received: 29 June 2019/Accepted: 6 December 2019

ABSTRACT

A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance.

Keywords: Discrete interference of chance; ergodic distribution; limit distribution; random walk; weak convergence

ABSTRAK

Proses jalan rawak semi-Markovan dengan kemungkinan gangguan umum telah dibangunkan dan dikaji. Isi utama kajian ini adalah andaian bahawa kemungkinan gangguan diskrit mempunyai bentuk umum. Dalam beberapa keadaan, terbukti bahawa prosesnya ergodik dan bentuk asal taburan ergodik serta fungsi pencirian prosesnya diperoleh. Dengan menggunakan identiti asas untuk jalan rawak, fungsi pencirian prosesnya diungkapkan oleh fungsi pencirian sempadan fungsian. Kemudian, pengembangan asimptotik dua penggal untuk fungsi pencirian piawai prosesnya ditemui. Dengan menggunakan pengembangan asimtotik ini, teorem penumpuan yang lemah untuk taburan ergodik daripada proses piawai dibuktikan dan bentuk pembatasan untuk taburan ergodik diperoleh. Taburan had yang diperoleh bertepatan dengan had taburan sisa masa menunggu proses pembaharuan yang dihasilkan oleh jujukan pemboleh ubah rawak yang mengungkapkan kemungkinan gangguan diskrit.

Kata kunci: Had taburan; jalan rawak; kemungkinan gangguan diskrit; penumpuan yang lemah; taburan ergodik

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*Corresponding author; email: ardicozlem@gmail.com