Necessary and sufficient condition for the Lamperti invariance principle. (Ukrainian, English) Zbl 1050.60040

Račkauskas, Alfredas; Suquet, Charles

Teor. Jmovirn. Mat. Stat. 68, 115-124 (2003); translation in Theory Probab. Math. Stat. 68, 127-138 (2004).

Let \(X_1,X_2,\ldots\) be i.i.d. mean zero random variables, \(S(t)=\sum_{k\leq t}X_{k}\), \(t>0\), \(S(0)=0\), \(\xi_{n}(t)=S([nt])+(nt-[nt])X_{[nt]+1}\), \(t\in[0,1]\). The main result of this paper is the following: Assume that \(0<\alpha<1/2\) and \(p=1/(1/2-\alpha)\). Then \(n^{-1/2}\xi_{n} @>\text{D}>> W\) in the Hölder space \(H_{\alpha}^0[0,1]\) if and only if \(\lim_{t\to\infty}t^{p}\text{P}(| X_1|\geq t)=0\). Here \(W(t)\) is a standard Wiener process and \(@>\text{D}>>\) denotes the convergence in distribution. The obtained results are used to investigate the change point problem under epidemic alternative.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)

Keywords:

necessary and sufficient condition; Lamperti invariance principle; Hölder space; weakly convergence