Why are random walks in dimensions 3 or higher transient? 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.

In 1 dimension, although the expected number of times that you will return to the origin before a give time approaches infinity as time approaches infinity, it varies sublinearly with time. When you have 3 independent 1-dimensional random walks all starting at the origin, although you can expect that each of them individually will return to the origin infinitely many times, it does not follow that there will ever be a time when all 3 of them are at the origin, although that might happen.
