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)].


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


[1] BILLINGSLEY, P. (1968). Convergence of Probability Mesures. Wiley, New York. · Zbl 0172.21201
[2] CSÖRG O, M., SZy SZKOWICZ, B. and WANG, Q. (2001). Donsker’s theorem and weighted approximations for self-normalized partial sums processes. Technical Report 360, Laboratory for Research in Statistics and Probability, Carleton Univ. and Univ. of Ottawa.
[3] CSÖRG O, M., SZy SZKOWICZ, B. and WANG, Q. (2003). Darling-Erd os theorems for selfnormalized sums. Ann. Probab. 31 676-692.
[4] EGOROV, V. A. (1996). On the asy mptotic behavior of self-normalized sums of random variables. Teor. Veroy atnost. i Primenen. 41 643-650.
[5] ERD OS, P. and KAC, M. (1946). On certain limit theorems of the theory of probability. Bull. Amer. Math. Soc. 52 292-302. · Zbl 0063.01274
[6] ERD OS, P. and KAC, M. (1947). On the number of positive sums of independent random variables. Bull. Amer. Math. Soc. 53 1011-1020. · Zbl 0032.03502
[7] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0138.10207
[8] GINE, E., GÖTZE, F. and MASON, D. M. (1997). When is the Student t-statistic asy mptotically standard normal? Ann. Probab. 25 1514-1531. · Zbl 0958.60023
[9] GRIFFIN, P. S. and KUELBS, J. D. (1989). Self-normalized laws of the iterated logatithm. Ann. Probab. 17 1571-1601. · Zbl 0687.60033
[10] GRIFFIN, P. S. and MASON, D. M. (1991). On the asy mptotic normality of self-normalized sums. Proc. Cambridge Phil. Soc. 109 597-610. · Zbl 0723.62008
[11] LÉVY, P. (1939). Sur certains processus stochastiques homogènes. Compositio Math. 7 283-339. · Zbl 0022.05903
[12] LOGAN, B. F., MALLOWS, C. L., RICE, S. O. and SHEPP, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 2 642-651. · Zbl 0272.60016
[13] O’BRIEN, G. L. (1980). A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab. 17 539-545. JSTOR: · Zbl 0428.60034
[14] PROHOROV, YU. V. (1956). Covergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157-214. · Zbl 0075.29001
[15] RA CKAUSKAS, A. and SUQUET, CH. (2000). Convergence of self-normalized partial sums processes in C[0, 1] and D[0, 1]. Pub. IRMA Lille 53-VI.
[16] RA CKAUSKAS, A. and SUQUET, CH. (2001). Invariance principles for adaptive self-normalized partial sums processes. Stochastic Process. Appl. 95 63-81. · Zbl 1059.60043
[17] SAKHANENKO, A. I. (1980). On unimprovable estimates of the rate of convergence in invariance principle. Colloquia Math. Soc. János Boly ai 32 779-783. · Zbl 0518.60046
[18] SAKHANENKO, A. I. (1984). On estimates of the rate of convergence in the invariance principle. In Advances in Probability Theory: Limit Theorems and Related Problems (A. A. Borovkov, ed.) 124-135. Springer, New York. · Zbl 0543.60040
[19] SAKHANENKO, A. I. (1985). Convergence rate in the invariance principle for non-identically distributed variables with exponential moments. In Advances in Probability Theory: Limit Theorems for Sums of Random Variables (A. A. Borovkov, ed.) 2-73. Springer, New York. · Zbl 0591.60027
[20] SPARRE-ANDERSEN, E. (1949). On the number of positive sums of random variables. Skand. Aktuarietidskrift 27-36. · Zbl 0041.45006
[21] TAKÁCS, L. (1981). The arc sine law of Paul Lévy. In Contributions to Probability. A Collection of Papers Dedicated to Eugene Lukács (J. Gani and V. K. Rohatgi, eds.). Academic Press, New York. · Zbl 0551.60069
[22] CANBERRA, ACT 0200 AUSTRALIA E-MAIL: qiying@wintermute.anu.edu.au
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