For a simple linear errors-in-variables regression model with widely orthant dependent errors, we provide sufficient conditions for the convergence rate in the strong consistency of the least squares estimators. We also provide necessary conditions. Our result improves and extends some results of Liu et al. (J. Math. Ineq., 14 (2020), 771-779).

Abstract. The Bernstein inequality is an exponential probability inequality for a sequence of bounded independent random variables. In this paper, we prove a Bernstein type inequality for unbounded negatively orthant dependent (NOD) random variables. As some applications, we obtain the convergence rates of the law of the iterated logarithm and law of the single logarithm for identically distributed NOD random variables. We also obtain a strong law for weighted sums of NOD random variables.Mathematics subject classification (2010): 60F15.

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The complete convergence result is obtained for weighted sums of ψ -mixing random variables without any conditions on mixing rate. As a special case, we can obtain the law of large numbers of Liu and Jin (J. Math. Ineq., 12, 2018). As applications, necessary and sufficient conditions are provided for the complete consistency of LS estimators in the errors-in-variables regression model with ψ -mixing errors.

Abstract. The Bernstein inequality is an exponential probability inequality for a sequence of bounded independent random variables. In this paper, we prove a Bernstein type inequality for unbounded negatively orthant dependent (NOD) random variables. As some applications, we obtain the convergence rates of the law of the iterated logarithm and law of the single logarithm for identically distributed NOD random variables. We also obtain a strong law for weighted sums of NOD random variables.Mathematics subject classification (2010): 60F15.