## 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

### Keywords:

Donsker’s theorem; self-normalized sums; arc sine law

### Citations:

Zbl 0063.01274; Zbl 0032.03502
Full Text:

### References:

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