Variance for exponential smoothing I want to obtain the analytical expression of variance for simple exponential smoothing . Please help verify and see if the expression could be further simplified , thanks .
Assume the discrete time process $(D_i)_{1 \le i\le t}$ is i.i.d , $(\hat{D}_i)_{1 \le i\le t}$ is the predictor .
The standard practice is to define the first smoothing factor as ,
$$ \hat{D}_1 =    D_1  $$
Let $T_a \ge 0 $ . Observe for t > 1 ,
$$  \hat{D}_t =  \left( \frac{1}{1+T_a} \right) D_t  + \left( 1 - \frac{1}{1+T_a} \right) \hat{D}_{t-1}$$
$$  \hat{D}_t = \left( \frac{1}{1+T_a} \right) D_t + \left( 1 - \frac{1}{1+T_a} \right)
\left(  \left( \frac{1}{1+T_a} \right) D_{t-1} +\left(  1 - \frac{1}{1+T_a}   \right)
 \left( ...      \right)    \right)$$
expanding out , and pulling out common term
$$ \hat{D}_t =  \left( \frac{1}{1+T_a} \right) \left[ D_t +  \left( 1 - \frac{1}{1+T_a}   \right) D_{t-1} +
 \left( 1 - \frac{1}{1+T_a}   \right)^2  D_{t-2} +  ... + 
\left( 1 - \frac{1}{1+T_a}    \right)^{t-2}  D_{t-(t-2)}   \right]  $$
$$ +   \left( 1 - \frac{1}{1+T_a}    \right)^{t-1} D_1 $$
Then , by i.i.d property
$$ Var(\hat{D_t}) = \left(\frac{1}{1+T_a} \right)^2
\left[   1 +   \left( 1 - \frac{1}{1+T_a}   \right)^2 +  \left( 1 - \frac{1}{1+T_a}   \right)^4 + ... +
 \left( 1 - \frac{1}{1+T_a}    \right)^{2(t-2)} \right] Var(D)
 $$
$$ +   \left( 1 - \frac{1}{1+T_a}    \right)^{2(t-1)} Var(D) $$
Since $ 0 \le 1 - \frac{1}{1+T_a}  < 1 $ , by formula for GP series
$$  Var(\hat{D_t})  =   \left\{   \left( \frac{1}{1+T_a} \right)^2
 \left[ \frac{1 - \left(1-  \frac{1}{1+T_a} \right)^{2  (t-1)}  }{1 -  \left( 1- \frac{1}{1+T_a} \right)^2  }  \right]
 +  \left( 1 - \frac{1}{1+T_a}   \right)^{2(t-1)}  \right\}  Var(D)  $$

Hi: Suppose that you have the following exponential smoothing model where the data, $y_t$, has variance $\sigma^2_y$ :
$\tilde y_t = (1-\lambda) \tilde y_{t-1} + \lambda y_t$.
Then it can be shown that $\sigma^2_{\tilde{y}_{t}} = \left(\frac{\lambda}{2-\lambda}\right)^2 \sigma^2_y$.
The proof is on page 72 of Box and Luceno's "Statistical Control by Monitoring and Feedback Adjustment".
