Probability for Statistics and Machine Learning pp 375-400 | Cite as

# Random Walks

## Abstract

We have already encountered the simple random walk a number of times in the previous chapters. Random walks occupy an extremely important place in probability because of their numerous applications, and because of their theoretical connections in suitable limiting paradigms to other important random processes in time. Random walks are used to model the value of stocks in economics, the movement of the molecules of a particle in a liquid medium, animal movements in ecology, diffusion of bacteria, movement of ions across cells, and numerous other processes that manifest random movement in time in response to some external stimuli. Random walks are indirectly of interest in various areas of statistics, such as sequential statistical analysis and testing of hypotheses. They also help a student of probability simply to understand randomness itself better.

## Keywords

Random Walk Independent Random Variable Simple Random Walk Asymptotic Distribution Theory General Random Walk## References

- Chung, K.L. (1974).
*A Course in Probability*, Academic Press, New York.MATHGoogle Scholar - Chung, K.L. (2001).
*A Course in Probability*, 3rd Edition, Academic Press, New York.Google Scholar - Chung, K.L. and Fuchs, W. (1951). On the distribution of values of sums of random variables,
*Mem. Amer. Math. Soc.*, 6, 12.MathSciNetGoogle Scholar - Dym, H. and McKean, H. (1972).
*Fourier Series and Integrals*, Academic Press, New York.MATHGoogle Scholar - Erd\ddot{
*o*}s, P. and Kac, M. (1947). On the number of positive sums of independent random variables,*Bull. Amer. Math. Soc.*, 53, 1011–1020.Google Scholar - Feller, W. (1971).
*Introduction to Probability Theory with Applications*, Wiley, New York.Google Scholar - Finch, S. (2003).
*Mathematical Constants*, Cambridge University Press, Cambridge, UK.MATHGoogle Scholar - Pólya, G. (1921). Uber eine Aufgabe der Wahrsch einlichkeitstheorie betreffend die Irrfahrt im Strasseenetz,
*Math. Annalen*, 84, 149–160.MATHCrossRefGoogle Scholar - Rényi, A. (1970).
*Probability Theory*, Nauka, Moscow.Google Scholar - Sparre-Andersen, E. (1949). On the number of positive sums of random variables,
*Aktuarietikskr*, 32, 27–36.Google Scholar - Spitzer, F. (2008).
*Principles of Random Walk*, Springer, New York.Google Scholar