Channel capacity with additive Gaussian noise with changing variance
I think you are looking for the information rate,

$$ I(x_{1},x_{2},\cdots,x_{n}; y_{1}, y_{2}, \cdots, y_{n}) = \frac{1}{n} \sum_{i=1}^{n} I( x_{i};y_{i} ) $$

when input achieves the capacity of the channel $ x \sim N(0,P) $. Then,

$$ C = \frac{1}{2n} \sum_{i=1}^{n} \log \left( 1 + \text{SNR}_{i} \right) $$

I think every time we use the channel we have a new random variable $ y $ since we have a memoryless channel and $ w $ changes. Then, we have a sequence of related random variables $ \left\lbrace x_{i} \right\rbrace $ and $ \left\lbrace y_{i} \right\rbrace $. With the information rate we compute the mutual information between this two sequences.

A good reference about this Elements of Information Theory.

I hope this could help you.