• Corpus ID: 252383384

Recurrence and transience of symmetric random walks with long-range jumps

  title={Recurrence and transience of symmetric random walks with long-range jumps},
  author={Johannes Baumler},
. Let X 1 , X 2 , . . . be i.i.d. random variables with values in Z d satisfying P ( X 1 = x ) = P ( X 1 = − x ) = Θ ( k x k − s ) for some s > d . We show that the random walk defined by S n = P nk =1 X k is recurrent for d ∈ { 1 , 2 } and s ≥ 2 d , and transient otherwise. This also shows that for an electric network in dimension d ∈ { 1 , 2 } the condition c { x,y } ≤ C k x − y k − 2 d implies recurrence, whereas c { x,y } ≥ c k x − y k − s for some c > 0 and s < 2 d implies transience. This… 

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  • C. Nash-Williams
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    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1959
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