Recurrent Random Walk

  • Frank Spitzer
Part of the Graduate Texts in Mathematics book series (GTM, volume 34)


The first few sections of this chapter are devoted to the study of aperiodic one-dimensional recurrent random walk.1 The results obtained in Chapter III for two-dimensional aperiodic recurrent random walk will serve admirably as a model. Indeed, every result in this section, which deals with the existence of the potential kernel a(x) = A(x,0), will be identically the same as the corresponding facts in Chapter III. We shall show that the existence of
$$a\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 0}^n {\left[ {{P_k}\left( {0,0} \right) - {P_k}\left( {x,0} \right)} \right]} $$
is a general theorem, valid for every recurrent random walk, under the very natural restriction that it be aperiodic. Only in section 29 shall we encounter differences between one and two dimensions. These differences become apparent when one investigates those aspects of the theory which depend on the asymptotic behavior of the potential kernel a(x) for large |x|. The result of P12.2, that
$$\mathop {\lim }\limits_{n \to \infty } \left[ {a\left( {x + y} \right) - a\left( x \right)} \right] = 0,y \in R$$
will be shown to be false in section 29 for aperiodic one-dimensional recurrent random walk with finite variance.


Random Walk Green Function Simple Random Walk Finite Variance Equilibrium Charge 
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Copyright information

© Frank Spitzer 1964

Authors and Affiliations

  • Frank Spitzer
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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