Let $x_t$ be a stationary process, such that autocorrelation dies down (is insignificant) after 3 lags. Let correlation at lag i be denoted by $\rho_i $
Let $y_t$ be given by $$y_t = \sum_{t-10}^t x_i .$$
Then, $$\sigma^2_y = \mathbb{V}(y_t) = \mathbb{V}\left(\sum_{t-10}^t x_i\right) \approx \sigma^2_x (10 + 2 * ( 9\rho_1 + 8 \rho_2 + 7 \rho_3 )),$$ where $\sigma^2_x = \mathbb{V}(x_1)$. Since $\sigma^2_y >0$, this gives a lower limit on $( 0.9\rho_1 + 0.8\rho_2 + 0.7\rho_3 ) $ of $-0.5$.
Is there a theoretical upper limit on $( 0.9\rho_1 + 0.8\rho_2 + 0.7\rho_3 )$ also?