On concentration of probability

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Distances in random graphs with infinite mean degrees (2004, methodology): “We will now use the Janson inequality [13], which states that for any t > 0,”
Distances in Random Graphs with Finite Mean and Infinite Variance Degrees (2005, methodology): “For the third term P(AN,ε(c)), we will use a bound given by Janson [24], which states that for a binomial random variable X with parameters N and p, and all t > 0, P(|X −Np| ≥ t) ≤ 2 exp { − t 2 2(Np + t/3) } .”