Complete answer to an interval splitting problem (Q1181119)

For \(n\geq 2\), let \(I_ k(n)\) denote the \(k\)-th largest interval of the \(n\) intervals generated by the splitting of \([0,1]\) through \(0,X_ 1,\dots,X_{n-1},1\), where \(X_ 1,X_ 2,\dots\) are independent uniformly distributed on \([0,1]\). The author proves that \(P(X_ n\in I_{k_ n}(n)\hbox{ i.o.})=1\) iff \(\sum_ n n^{-1}\log(n/k_ n)=\infty\).