## Abstract

Let X, X_{1}, X_{2},... be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if max_{1≤j≤n} |X_{j}|/V_{n} → _{P} 0, as n → ∞, where V_{n}^{2} = ∑_{j=1}^{n} X_{j}^{2}.

Original language | English |
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Pages (from-to) | 1228-1240 |

Number of pages | 13 |

Journal | Annals of Probability |

Volume | 31 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2003 |