I watched this PBS video a while ago (relevant part here) and have been trying to get my head around the idea of transient walks. The video says that a recurrent random walk is one that is guaranteed to return to it's starting position - all 1D and 2D walks - and a walk is transient if there is a positive probability that it never returns - 3D or higher. I've tried to have a think about this and looked some stuff up but I haven't had any breakthroughs.
What confuses me is this: A random walk in 3 dimensions can be split up into 3 independent random 1D walks. If each of these walks is guaranteed to return to the starting position infinitely many times we can say that there is a finite positive probability that they will return to the starting point on a given 'turn'. The product of the three finite probabilities is finite so isn't there a finite chance that any random walk in three dimensions will return to the start on any given 'turn' and hence they are guaranteed to return at some point?
I imagine I am just making incorrect assumptions about the nature of these infinite systems as is too easy to do but I'd like to know exactly where my intuition is wrong.