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 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 444 Thanks: 29 Math Focus: Number theory  
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?

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 1D or 2D lattices but not 3D or higher. 
August 12th, 2019, 11:18 PM  #6  
Senior Member Joined: Oct 2009 Posts: 850 Thanks: 325  Quote:
 

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