The bootstrap can be used to generate a new random sample from an existing random sample. Its validity can be guaranteed by the Glivenko-Cantelli theorem, which demonstrates how the empirical cumulative distribution (CDF, top panel), converges on the CDF of the sample (bottom panel).

For an increasing function 𝑓:ℝ→ℝ, max(𝑓(𝑎),𝑓(𝑏))=𝑓(max(𝑎,𝑏)). An important special case is 𝑓(𝑥)=𝑥+𝑐, for which we obtain max(𝑎+𝑐,𝑏+𝑐)=𝑐+max(𝑎,𝑏).