Hájek-rényi Inequality for Dependent Random Variables in Hilbert Space and Applications
نویسندگان
چکیده
It is well-known that Kolmogorov’s inequality is the particular case bk = 1, for all k and n = 1 in (1.1). Afterwards this inequality was extended to real valued martingales (see [4]). Since then, this inequality has been studied by many authors. For the case of R-valued random variables, Sung [22] obtained the Hájek-Rényi inequality for the associated sequence. Liu et al. [19] considered the negatively associated random variables. Cohn [5] studied a Hájek-Rényi inequality for Markov chain. Tómács and Ĺıbor [23], Hu et al. [12] showed the inequality for demimartingale. For the case of Banach space, Gan [8] gave the Hájek-Rényi inequality for martingale. Furthermore, Gan and Qiu [9] studied a general version of this inequality. In this paper, we extend the Hájek-Rényi inequality of associated, negatively associated and φ-mixing for random sequences to a separable real Hilbert space and derive the strong law of large numbers for these dependent sequences with values in a separable real Hilbert space.
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