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Bernstein inequality for the parameter of the pth order autoregressive process AR(p)

Samir Benaissa (2006)

Applicationes Mathematicae

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: X ̃ t = θ ̃ X ̃ t - 1 + ε ̃ t . In this paper we study the convergence in distribution of the linear operator I ( θ ̃ T , θ ̃ ) = ( θ ̃ T - θ ̃ ) θ ̃ T - 2 for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.

Complete convergence in mean for double arrays of random variables with values in Banach spaces

Ta Cong Son, Dang Hung Thang, Le Van Dung (2014)

Applications of Mathematics

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p ). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.

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