I would like to calculate the characteristic function of
$Z_{\beta, n}=(1-\beta^2)^{1/2}\sum_{k=0}^n\beta^kX_k$,
where $X_i$ are independent random variables with $P(X_i = 1)=P(X_i=-1) = 1/2$ and $\beta \in (0,1)$.
The formula would be $\phi(t) = \int_\mathbb{R}e^{itZ_{\beta,n}(x)}p_{Z_{\beta,n}}(x)dx$ where $p_{Z_{\beta,n}}$ is the density of $Z_{\beta,n}$.
I am not sure how exactly the density looks like and how to calculate the complete integral.
I don't know how useful my final answer here is, but by independence, one can write the characteristic function directly as
$$ \phi (t ) = E \,e^{it Z} = \prod_{k=0}^nE\, e^{is\beta^kX_k},$$ where $s = t (1-\beta^2)^{1/2}$.
One can evaluate each term of the product explicitly: $$ E\,e^{is\, \beta^kX_k} = 1/2\, (e^{is\beta^k} + e^{-is\beta^k} ) = \cos (\,s\beta^k\, ). $$ Therefore:
$$ \phi(t ) = \prod_{k=0}^n \cos(\, s \beta^k\,).$$