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Donsker’s theorem for self-normalized partial sums processes. (English) Zbl 1045.60020

Let \(X,X_1,X_2,\dots\) be a sequence of nondegenerate i.i.d. random variables and let \[ S_n= \sum^n_{j=1} X_j,\quad V^2_n= \sum^n_{j=1} X_j\quad (n= 1,2,\dots). \] The authors prove the following self-normalized version of the weak invariance principle. As \(n\to\infty\), the following statements are equivalent:
(a) \(EX= 0\) and \(X\) is in the domain of attraction of the normal law;
(b) \(S_{[nt_0]}/V_n @>D>> N(0,t_0)\) for \(t_0\in (0,1]\);
(c) \(S_{[nt]}/V_n @>D>> W(t)\) on \((D[0,1],\rho)\), where \(\rho\) is the sup-norm metric for functions in \(D[0,1]\) and \(\{W(t):0\leq t\leq 1\}\) is a standard Wiener process;
(d) on an appropriate probability space for \(X,X_1,X_2,\dots\), we can construct a standard Wiener process \(\{W(t):0\leq t<\infty\}\) such that \[ \sup_{0\leq t\leq 1}\,| S_{[nt]}/V_n- W(nt)/\sqrt{n}|= o_P(1). \] Further, the authors obtain that if \(X\) is symmetric around mean zero, then \[ \max_{1\leq j\leq n}\,| X_j|/V_n @>P>> 0\Leftrightarrow S_{\widetilde K_n(t)}/ V_n @>D>> W(t)\quad (D[0,1],\rho), \] where \(\widetilde K(t)= \sup\{mV^2_m\leq tV^2_n\}\).
By the former result we can easily extend one by P. Erdős and M. Kac [Bull. Am. Math. Soc. 52, 292–302 (1946)], and the latter result deduces a general form of Lévy’s arcsine law [P. Erdős and M. Kac, ibid. 53, 1011–1020 (1947; Zbl 0032.03502)].

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
62E20 Asymptotic distribution theory in statistics
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References:

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[22] CANBERRA, ACT 0200 AUSTRALIA E-MAIL: qiying@wintermute.anu.edu.au
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