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: $X{̃}_t=\theta ̃X{̃}_{t-1}+\epsilon {̃}_t$. In this paper we study the convergence in distribution of the linear operator $I(\theta {̃}_T,\theta ̃)=(\theta {̃}_T-\theta ̃)\theta {̃}^{T-2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.