Renyi's lemma and some of its applications

One of the most important theorems of the theory of discrete probability, but unfortunately among the lesser known, is Renyi's lemma. Renyi's lemma, roughly speaking, says that a certain relation will hold if and only if it holds for all combinations of the ‘sure’ and ‘null’ events. The applications of Renyi's lemma are plentiful. The simplest one is a very short proof of the principle of inclusion‐exclusion. Another application is to the theory of distributions. More specifically, it is known that the sum of independent identically distributed Bernoulli random variables is distributed according to the Binomial law. It is much harder, however, to determine the distribution of the sum of non‐independent Bernoulli random variables. Renyi's lemma proves useful in as much as it fills the gap between the independent and the dependent cases. Another application of Renyi's lemma is to the theory of random graphs. It is a surprising fact that under only mild conditions the random graph behaves (asymptotically) like many natural phenomena, in the sense that many counting processes are distributed according to the Poisson law.