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 August 12th, 2019, 04:22 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 444 Thanks: 29 Math Focus: Number theory Random walk returns home How does one show that a random walk eventually returns to its origin?
August 12th, 2019, 04:36 PM   #2
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Quote:
 Originally Posted by Loren How does one show that a random walk eventually returns to its origin?
Doesn't this condition approach its whole map to be covered eventually?

 August 12th, 2019, 04:47 PM #3 Senior Member   Joined: Jun 2019 From: USA Posts: 120 Thanks: 40 If the probability density function of the step size (in however many dimensions we are considering) is exactly symmetric about zero, then yes, after an infinite number of steps, it should eventually reach every value. On the other hand, if you allow for infinite precision, then the probability of a sum of truly random numbers exactly equalling zero is, well, zero. Are we sure it returns to the origin? Thanks from Loren
 August 12th, 2019, 04:58 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,921 Thanks: 2203 Are you referring to PĆ³lya's recurrence theorem?
 August 12th, 2019, 05:39 PM #5 Senior Member   Joined: Jun 2019 From: USA Posts: 120 Thanks: 40 I tell you, nothing makes you Google more things than listening to mathematicians talk about maths. Yes, if the walk is of equidistant steps on a regular lattice, that certainly changes things. 2013 short paper by Mare (first Google search result) shows one proof, and why it's apparently true for 1-D or 2-D lattices but not 3-D or higher.
August 12th, 2019, 11:18 PM   #6
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Quote:
 Originally Posted by DarnItJimImAnEngineer If the probability density function of the step size (in however many dimensions we are considering) is exactly symmetric about zero, then yes, after an infinite number of steps, it should eventually reach every value. On the other hand, if you allow for infinite precision, then the probability of a sum of truly random numbers exactly equalling zero is, well, zero. Are we sure it returns to the origin?
Yes, you never return to the origin with infinite precision, but you can get arbitrarily close. Given any distance from the origin, you will eventually get closer than that to the origin.

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