J. Klafter and I.M. Sokolov
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0001
- Subject:
- Physics, Soft Matter / Biological Physics
Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter ...
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Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter introduces the necessary mathematical instruments, and then use them to discuss general expressions for the distribution of the walker's displacement after a given number of steps in one dimension and in higher dimensions. It moreover discusses moments of displacement, provided these moments exist. The chapter then considers a simple approach to the central limit theorem, and discusses situations, when this breaks down (corresponding to the cases when the second moment of step lengths diverges).Less
Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter introduces the necessary mathematical instruments, and then use them to discuss general expressions for the distribution of the walker's displacement after a given number of steps in one dimension and in higher dimensions. It moreover discusses moments of displacement, provided these moments exist. The chapter then considers a simple approach to the central limit theorem, and discusses situations, when this breaks down (corresponding to the cases when the second moment of step lengths diverges).
