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Random Walk

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International Encyclopedia of Statistical Science

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The simple random walk {S n : n = 0, 1, …}, starting at an integerx, is a stochastic process on the integers, given by S 0 = x, S n = x + X 1 + … + X n (n ≥ 1), where X n , n ≥ 1,is an independent Bernoulli sequence:P(X n = 1) = p, P(X n = − 1) = 1 − p = q, 0 < p < 1. In the case, p = q = 1 ∕ 2, it is called the simple symmetric random walk, while if p≠1 ∕ 2, it is asymmetric. By the binomial theorem, \(P({S}_{n} = y\mid {S}_{0} = 0) = {C}_{\frac{(n+y)} {2} }^{n}{p}^{(n+y)/2}{q}^{(n-y)/2}\), if y and n are of the same parity, i.e., if either both are odd or both are even. Otherwise, P(S n = y∣S 0 = 0) = 0. Here = C m n = n! ∕ (m! (n − m)! ).

For c ≤ x ≤ d integers, the probability π(x) that a simple random walk, starting at x, reaches c before d satisfies the equation

$$\begin{array}{rcl} \pi (x)& =& p\pi (x + 1) + q\pi (x - 1) \\ & & \text{ for $c <x <d$, $\pi (c) = 1$, $\pi (d) = 0$,} \\ \end{array}$$
(1)

as shown by conditioning on the first step X 1. For the symmetric walk, the...

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References and Further Reading

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  • Feller W (1968) An introduction to probability theory and its applications. vol 1. 3rd edn. Wiley, New York

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  • Lévy P (1925) Calcul des probabilités. Gauthier-Villars, Paris

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  • Spitzer F (1964) Principles of random walk. Van Nostrand. Princeton

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Author information

Authors and Affiliations

  1. The University of Arizona, Tucson, AZ, USA

    Rabi Bhattacharya (Professor of Mathematics)

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  1. Rabi Bhattacharya
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Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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Bhattacharya, R. (2011). Random Walk. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_475

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