Necessary conditions for the bootstrap of the mean of a triangular array

Although necessary conditions for the bootstrap of the mean to work have al ready been given, only the case of an i.i.d. sequence has been exhaustively considered. We study such necessary conditions for (not necessarily infini tesimal) triangular arrays showing that the existence of a limit law in pro bability leads to infinitesimality and to the Central Limit Theorem to hold for a rescaled subarray. Our setup is based on a triangular array of row-wise independent identicall y distributed random variables and any resampling size. While our results a re similar to those obtained in Arcones and Gine [Ann. Inst. Henri Poincare 25(1989) 457-481] for an i.i.d. sequence, our proof is based on symmetriza tions and the consideration of U-statistics and allows a unified treatment without moment assumptions. (C) Elsevier, Paris.