Let $\{x_n\}$ be a sequence of positive numbers and $\alpha \in (0,1)$. Let $y_0 := x_n$ and $$ y_k := (\alpha \,y_{k-1} + 1-\alpha) x_{n-k} $$ for $k=1,2,\dots,n-1$.
Is it possible to give a sharp bound on $y_{n-1}$ based on say $\prod_{i=1}^{n} x_i$?