Introduction

In 1979, Efron introduced the bootstrap method as a kind of universal tool to obtain approximation of the distribution of statistics. The now well known underlying idea is the following: consider a sample X₁,…, X_n of independent and identically distributed (i.i.d.) random variables (r.v.’s) with unknown probability measure (p.m.) P. Assume we are interested in approximating the distribution of a statistical functional T(P_n), the empirical counterpart of the functional T(P), where $$ {P_n}: = {n^{{ - 1}}}\Sigma_{{i = 1}}^n{\delta_{{{X_i}}}} $$ is the empirical p.m. Since in some sense P_n is close to P when n is large, if one samples X₁*,…, $$ {X_{{{m_n}}}}^{ * } $$ i.i.d. from P_n and builds the empirical p.m. $$ {P^{ * }}_{{{m_n}}}: = m_n^{{ - 1}}\Sigma_{{i = 1}}^{{{m_n}}}{\delta_{{{X_i}}}} * $$, then the behaviour of $$ T\left( {{P^{ * }}_{{n,{m_n}}}} \right) $$ conditionally on P_n should imitate that of T(P_n) when n and m_n get large.