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Let $$ x_t =\sum_j \alpha_j v_{t-j}$$ and $$y_t =\sum_j \beta_j w_{t-j}$$ are two linear processes. Let $v_i \sim \mathcal{N}(0,\sigma_v^2)$ and $w_i \sim \mathcal{N}(0,\sigma_w^2)$ are two white noise sequences independent of each other. What is the asymptotic distribution of the correlation coefficient $\rho_{xy}$ ?

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