Weighted Sum of Auto Correlation - Lower Bound
Defining $ a \left[ k \right] = {2}^{- \left| k \right|} $.
Moreover, the Auto Correlation function of $ v $ defined as $ {r}_{vv} \left[ k \right] = \left \langle {v}^{\left( 0 \right)}, {v}^{\left( k \right)} \right \rangle = \sum_{n = -\infty}^{\infty} {v}_{n} {v}_{n - k} $.
Pay attention that Auto Correlation is Hermitian Function.

Using the definition of Convolution one could write:

$$ \left( {r}_{vv} \ast a \right) \left[ 0 \right] = \sum_{k = -\infty}^{\infty} {2}^{- \left| k \right| } \left \langle {v}^{\left( 0 \right)}, {v}^{\left( k \right)} \right \rangle $$

Using the Convolution Theorem one could write that:

$$ \left( {r}_{vv} \ast a \right) \left[ 0 \right] = \int_{- \pi}^{\pi} {R}_{vv} \left( \omega \right) A \left( \omega \right) d \omega $$

Where $ R \left( \omega \right) $ and $ {R}_{vv} \left( \omega \right) $ are the DTFT of $ {r}_{vv} \left[ k \right] $ and $ a \left[ k \right] $ respectively. 

One should notice the DTFT of $ a \left[ k \right] $ is defined only one sided. Yet since its symmetrical it can well calculated:

$$
\begin{align*}
A \left( \omega \right) &amp; = DTFT \left\{ a \left[ k \right]  \right\} = \sum_{k = -\infty}^{\infty} a \left[ k \right] {e}^{-j \omega k} = \sum_{k = 0}^{\infty} {2}^{-k} {e}^{-j \omega k} + \sum_{k = 0}^{\infty} {2}^{-k} {e}^{j \omega k} - 1 \\
&amp; = \frac{1}{1 - 0.5 {e}^{-j \omega}} + \frac{1}{1 - 0.5 {e}^{j \omega}} - 1 = \frac{1 - {c}^{2}}{1 - 2 c \cos \left( \omega \right) + {c}^{2}} = \alpha &gt; 0 \quad \forall c &lt; 1
\end{align*}
$$

In the above $ c = {2}^{-1} = 0.5 $ yet actually this will hold for any $ c &lt; 1 $.

So the integral is given by:

$$
\begin{align*}
\int_{- \pi}^{\pi} {R}_{vv} \left( \omega \right) A \left( \omega \right) d \omega &amp; = \int_{- \pi}^{\pi} {R}_{vv} \left( \omega \right) \frac{1 - {c}^{2}}{1 - 2 \alpha \cos \left( \omega \right) + {c}^{2}} d \omega \\
&amp; \geq \alpha \int_{- \pi}^{\pi} {R}_{vv} \left( \omega \right) d \omega = \alpha {\left\| v \right\|}^{2}
\end{align*}
$$

As requested.

By the way the result must be real since $ a \left[ k \right] $ is symmetric and $ {r}_{vv} $ is hermitian function and hence its transform is real.