IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i21p4513-d1272525.html
   My bibliography  Save this article

On the Height of One-Dimensional Random Walk

Author

Listed:
  • Mohamed Abdelkader

    (Department of Statistics and Operations Research, Faculty of Sciences, King Saud University, Riyadh 11451, Saudi Arabia)

Abstract

Consider the one-dimensional random walk X n : as it evolves (at each unit of time), it either increases by one with probability p or resets to 0 with probability 1 − p . In the present paper, we analyze the law of the height statistics H n , corresponding to our model X n . Also, we prove that the limiting distribution of the walk X n is a shifted geometric distribution with parameter 1 − p and find the closed forms of the mean and the variance of X n using the probability-generating function.

Suggested Citation

  • Mohamed Abdelkader, 2023. "On the Height of One-Dimensional Random Walk," Mathematics, MDPI, vol. 11(21), pages 1-12, November.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4513-:d:1272525
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/21/4513/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/21/4513/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Endre Csáki & Yueyun Hu, 2001. "Asymptotic Properties of Ranked Heights in Brownian Excursions," Journal of Theoretical Probability, Springer, vol. 14(1), pages 77-96, January.
    2. Kankal, Murat & AkpInar, Adem & Kömürcü, Murat Ihsan & Özsahin, Talat Sükrü, 2011. "Modeling and forecasting of Turkey's energy consumption using socio-economic and demographic variables," Applied Energy, Elsevier, vol. 88(5), pages 1927-1939, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Leung, Philip C.M. & Lee, Eric W.M., 2013. "Estimation of electrical power consumption in subway station design by intelligent approach," Applied Energy, Elsevier, vol. 101(C), pages 634-643.
    2. Santika, Wayan G. & Anisuzzaman, M. & Simsek, Yeliz & Bahri, Parisa A. & Shafiullah, G.M. & Urmee, Tania, 2020. "Implications of the Sustainable Development Goals on national energy demand: The case of Indonesia," Energy, Elsevier, vol. 196(C).
    3. Pesantez, Jorge E. & Li, Binbin & Lee, Christopher & Zhao, Zhizhen & Butala, Mark & Stillwell, Ashlynn S., 2023. "A Comparison Study of Predictive Models for Electricity Demand in a Diverse Urban Environment," Energy, Elsevier, vol. 283(C).
    4. Endre Csáki & Miklós Csörgő & Antónia Földes & Pál Révész, 2016. "Some Limit Theorems for Heights of Random Walks on a Spider," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1685-1709, December.
    5. Syed Aziz Ur Rehman & Yanpeng Cai & Rizwan Fazal & Gordhan Das Walasai & Nayyar Hussain Mirjat, 2017. "An Integrated Modeling Approach for Forecasting Long-Term Energy Demand in Pakistan," Energies, MDPI, vol. 10(11), pages 1-23, November.
    6. Vu, D.H. & Muttaqi, K.M. & Agalgaonkar, A.P., 2015. "A variance inflation factor and backward elimination based robust regression model for forecasting monthly electricity demand using climatic variables," Applied Energy, Elsevier, vol. 140(C), pages 385-394.
    7. Bilgili, Mehmet & Pinar, Engin, 2023. "Gross electricity consumption forecasting using LSTM and SARIMA approaches: A case study of Türkiye," Energy, Elsevier, vol. 284(C).
    8. Zhang, Jinjun & Abbasi, Kashif Raza & Hussain, Khadim & Akram, Sabahat & Alvarado, Rafael & Almulhim, Abdulaziz I., 2022. "Another perspective towards energy consumption factors in Pakistan: Fresh policy insights from novel methodological framework," Energy, Elsevier, vol. 249(C).
    9. Uzlu, Ergun & Akpınar, Adem & Özturk, Hasan Tahsin & Nacar, Sinan & Kankal, Murat, 2014. "Estimates of hydroelectric generation using neural networks with the artificial bee colony algorithm for Turkey," Energy, Elsevier, vol. 69(C), pages 638-647.
    10. Mehmet Kayakuş, 2020. "The Estimation of Turkey's Energy Demand Through Artificial Neural Networks and Support Vector Regression Methods," Alphanumeric Journal, Bahadir Fatih Yildirim, vol. 8(2), pages 227-236, December.
    11. Laimon, Mohamd & Mai, Thanh & Goh, Steven & Yusaf, Talal, 2022. "System dynamics modelling to assess the impact of renewable energy systems and energy efficiency on the performance of the energy sector," Renewable Energy, Elsevier, vol. 193(C), pages 1041-1048.
    12. Schlembach, Christoph & Schmidt, Sascha L. & Schreyer, Dominik & Wunderlich, Linus, 2022. "Forecasting the Olympic medal distribution – A socioeconomic machine learning model," Technological Forecasting and Social Change, Elsevier, vol. 175(C).
    13. Aydin, Gokhan, 2014. "Modeling of energy consumption based on economic and demographic factors: The case of Turkey with projections," Renewable and Sustainable Energy Reviews, Elsevier, vol. 35(C), pages 382-389.
    14. Emre Yakut & Ezel Özkan, 2020. "Modeling of Energy Consumption Forecast with Economic Indicators Using Particle Swarm Optimization and Genetic Algorithm: An Application in Turkey between 1979 and 2050," Alphanumeric Journal, Bahadir Fatih Yildirim, vol. 8(1), pages 59-78, June.
    15. Yuehjen E. Shao & Yi-Shan Tsai, 2018. "Electricity Sales Forecasting Using Hybrid Autoregressive Integrated Moving Average and Soft Computing Approaches in the Absence of Explanatory Variables," Energies, MDPI, vol. 11(7), pages 1-22, July.
    16. Colmenar, J.M. & Hidalgo, J.I. & Salcedo-Sanz, S., 2018. "Automatic generation of models for energy demand estimation using Grammatical Evolution," Energy, Elsevier, vol. 164(C), pages 183-193.
    17. Fazle Wahid & Hamid Ullah & Sher Ali & Sajjad Ahmad Jan & Abid Ali & Azhar Khan & Imran Ali Khan & Maryam Bibi, 2021. "The Determinants and Forecasting of Electricity Consumption in Pakistan," International Journal of Energy Economics and Policy, Econjournals, vol. 11(1), pages 241-248.
    18. Jeong, Kwangbok & Koo, Choongwan & Hong, Taehoon, 2014. "An estimation model for determining the annual energy cost budget in educational facilities using SARIMA (seasonal autoregressive integrated moving average) and ANN (artificial neural network)," Energy, Elsevier, vol. 71(C), pages 71-79.
    19. Salari, Mahmoud & Javid, Roxana J., 2017. "Modeling household energy expenditure in the United States," Renewable and Sustainable Energy Reviews, Elsevier, vol. 69(C), pages 822-832.
    20. Wang, Xingwei & Cai, Yanpeng & Chen, Jiajun & Dai, Chao, 2013. "A grey-forecasting interval-parameter mixed-integer programming approach for integrated electric-environmental management–A case study of Beijing," Energy, Elsevier, vol. 63(C), pages 334-344.

    More about this item

    Keywords

    height; return time; random walk;
    All these keywords.

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4513-:d:1272525. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.