Definition of a system of recurrent events

[I asked a version of this question on MSE a few weeks ago and didn’t get any useful feedback. Apologies if I am just being stupid.]

I am reading the paper A note on the Borel-Cantelli lemma by Kochen and Stone. They say:

A sequence $E_1, E_2, dotsc$ is called a system of recurrent events if there exist independent and identically distributed positive-integer valued random variables $Y_1, Y_2, dotsc$ such that $E_k$ is the event that $Y_1 + dotsb + Y_j = k$ for some $j$.

I note that this seems to be stronger than the definition given in Feller, say, but let’s bear with it. Later on they give an example:

Let $E_k$ be the event that the simple random walk in one dimension is at the origin at time $2k$. The $E_k$ form a system of recurrent events ….

This is a recurrent system per Feller’s definition, but I can’t reconcile it with Kochen and Stone’s definition. What could the sequence $Y_1, Y_2, dotsc$ be in the example?