Let {X,Xn; n≧0} be a sequence of identically distributed ψ-mixing dependent random variables taking values in a type 2 Banach space B with topological dual B∗. Considering the geometrically weighted series for 0<β<1, a general law of the iterated logarithm for ξ(β) is obtained without second moment.
[1] Berkes, I., Philipp, W. 1979 Approximation theorems for independent and weakly dependent random vectors Ann. Probab. 7 29–54 .
[2] Bovier, A., Picco, P. 1993 A law of the iterated logarithm for random geometric series Ann. Probab. 21 168–184 .
[3] Dehling, H., Philipp, W. 1982 Almost sure invariance principles for weakly dependent vector-valued random variables Ann. Probab. 10 689–701 .
[4] Einmahl, U., Li, D. L. 2005 Some results on two-sided LIL behavior Ann. Probab. 33 1601–1624 .
[5] Einmahl, U., Li, D. L. 2008 Characterization of LIL behavior in Banach space Trans. Amer. Math. Soc. 360 6677–6693 .
[6] Fu, K. A. 2010 LIL behavior for weakly dependent random variables in Banach spaces Acta Math. Hungar. 128 315–327 .
[7] Huang, W. 2003 A law of the iterated logarithm for geometrically weighted series of negatively associated random variables Statist. Probab. Lett. 63 133–143 .
[8] Kuelbs, J., Philipp, W. 1980 Almost sure invariance principles for partial sums of mixing B-valued random variables Ann. Probab. 8 1003–1036 .
[9] Peligrad, M. 1982 Invariance principles for mixing sequences of random variables Ann. Probab. 10 968–981 .
[10] Philipp, W., Stout, W. 1975 Almost sure invariance principles for partial sums of weakly dependent random variables Memoirs of Amer. Math. Soc. 2 161.
[11] Shao, Q. M. 1993 Almost sure invariance principles for mixing sequences of random variables Stoch. Proc. Appl. 48 319–334 .
[12] Picco, P., Vares, M. E. 1994 A law of the iterated logarithm for geometrically weighted martingale difference sequences J. Theor. Probab. 7 375–415 .
[13] Sharipov, O. Sh. 1998 Probabilistic characterization of Banach spaces by weakly dependent random elements Uzbek. Mat. Zh. 2 86–91 (in Russian).
[14] Sharipov, O. Sh. 2003 The law of the iterated logarithm for weakly dependent Hilbert space valued random variables Sib. Math. J. 44 1111–1126 .
[15] Sharipov, O. Sh. 2005 The bounded law of the iterated logarithm for weakly dependent random variables with values in a type 2 Banach space Theor. Probab. Appl. 49 444–457 .
[16] Zhang, L. X. 1997 Strong approximation theorems for geometrically weighted random series and their applications Ann. Probab. 25 1621–1635 .
[17] Zhang, L. X. 1997 A law of the iterated logarithm for geometrically weighted series of B-valued random variables Acta Sci. Math. (Szeged) 63 671–688.