Introduction to Random Processes

1. Front Matter

   Pages I-VIII

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2. Random Processes with Discrete State Space

   • Yuriĭ A. Rozanov

   Pages 1-9

3. Homogeneous Markov Processes with a Countable Number of States

   • Yuriĭ A. Rozanov

   Pages 10-17

4. Homogeneous Markov Processes with a Countable Number of States

   • Yuriĭ A. Rozanov

   Pages 18-24

5. Branching Processes

   • Yuriĭ A. Rozanov

   Pages 25-32

6. Brownian Motion

   • Yuriĭ A. Rozanov

   Pages 33-43

7. Random Processes in Multi-Server Systems

   • Yuriĭ A. Rozanov

   Pages 44-51

8. Random Processes as Functions in Hilbert Space

   • Yuriĭ A. Rozanov

   Pages 52-56

9. Stochastic Measures and Integrals

   • Yuriĭ A. Rozanov

   Pages 57-60

10. The Stochastic Ito Integral and Stochastic Differentials

    • Yuriĭ A. Rozanov

    Pages 61-67

11. Stochastic Differential Equations

    • Yuriĭ A. Rozanov

    Pages 68-72

12. Diffusion Processes

    • Yuriĭ A. Rozanov

    Pages 73-76

13. Linear Stochastic Differential Equations and Linear Random Processes

    • Yuriĭ A. Rozanov

    Pages 77-83

14. Stationary Processes

    • Yuriĭ A. Rozanov

    Pages 84-91

15. Some Problems of Optimal Estimation

    • Yuriĭ A. Rozanov

    Pages 92-99

16. A Filtration Problem

    • Yuriĭ A. Rozanov

    Pages 100-117

17. Back Matter

    Pages 108-120

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Today, the theory of random processes represents a large field of mathematics with many different branches, and the task of choosing topics for a brief introduction to this theory is far from being simple. This introduction to the theory of random processes uses mathematical models that are simple, but have some importance for applications. We consider different processes, whose development in time depends on some random factors. The fundamental problem can be briefly circumscribed in the following way: given some relatively simple characteristics of a process, compute the probability of another event which may be very complicated; or estimate a random variable which is related to the behaviour of the process. The models that we consider are chosen in such a way that it is possible to discuss the different methods of the theory of random processes by referring to these models. The book starts with a treatment of homogeneous Markov processes with a countable number of states. The main topic is the ergodic theorem, the method of Kolmogorov's differential equations (Secs. 1-4) and the Brownian motion process, the connecting link being the transition from Kolmogorov's differential-difference equations for random walk to a limit diffusion equation (Sec. 5).

• Brownian motion
• Markov process
• Random variable
• diffusion process
• ergodic theory
• filtration
• random walk
• stochastic differential equation

• Steklov Mathematical Institute, Moscow, USSR

  Yuriĭ A. Rozanov

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