[I asked a [version](https://math.stackexchange.com/questions/3747401/definition-of-a-system-of-recurrent-events) 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](http://projecteuclid.org/euclid.ijm/1256059668). They say:



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> *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 $j$.*
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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:



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> *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 ….*
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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?



Definition of a system of recurrent events
Assume the random walk starts at the origin. Let $T\_0=0$ and for $i\geq 1$ let $T\_i$ be the time of the $i$th visit of the walk to the origin after time $0$. Let $X\_i=T\_i-T\_{i-1}$. Then the $X\_i$ are i.i.d. (you may like to think of this in terms of the strong Markov property, for example), and they take even positive integer values.


Let $Y\_i=X\_i/2$, so that $Y\_i$ are i.i.d. taking positive integer values.The random walk is at $0$ at time $2k$ precisely if $T\_j=2k$ for some $j$, i.e. if $X\_1+\dots+X\_j=2k$, i.e. if $Y\_1+\dots+Y\_j=k$.


