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The Borel-Cantelli lemmas turn a series of event probabilities into a statement about an entire infinite sequence of observations. A summable sequence of probabilities forces only finitely many occurrences almost surely, regardless of independence. A divergent series forces infinitely many occurrences when the events are mutually independent events. The precise event being measured is the limsup of events.

Imagine alarms with probabilities $p_n=1/(n+1{)}^2$ or $p_n=1/(n+1)$, starting at $n=1$. Both individual probabilities tend to zero. Nevertheless, the first alarm sequence eventually stops almost surely, whereas the second keeps returning almost surely if its events are mutually independent events. Reusing one uniform random variable for every alarm instead produces the shared-uniform counterexample to the second Borel-Cantelli lemma. Thus the individual probabilities alone do not determine long-run recurrence.

Decreasing strictly positive summable sequences with equal totals and $x\prec y$ admit an infinite doubly stochastic matrix realizing the displayed relation. Choose $k$ with $y_k\ge x_1>y_{k+1}$ and average the pair to obtain $x_1$ and $y_k+y_{k+1}-x_1$. The remaining decreasing sequence majorizes the remaining target sequence: short initial sums contain values at least $x_1$, and longer sums are the old sums minus $x_1$. Repeat on the tail, freezing one normalized finite-support row at each stage. Limit column sums are at most one; Tonelli's theorem, equal finite totals and strict positivity of every $y_j$ force every column sum to equal one.

All sequence spaces here are over $\mathbb{C}$. Use the supremum norm on ${\ell }^{\infty },c_0,c_{00}$ and the summable sequence norm $\Vert a{\Vert }_1={\sum }_j|a_j|$ on ${\ell }^1$. The dual consists of continuous complex-linear functionals.