Definition (Discrete-Time Continuous Memoryless Channels)

Consider a discrete-time channel with continuous input and output alphabets given by $\mathcal{X}\subset\mathbb{R}$ and $\mathcal{Y}\subset\mathbb{R}$ respectively, and described by a sequence of $n$-fold conditional pdfs $f_{Y^{n}|X^{n}}$ which govern receiving $Y^{n}=(Y_{1},\ldots,Y_{n})$ at channel output when $X^{n}=(X_{1},\ldots,X_{n})$ is sent at the input $$(\mathcal{X},\mathcal{Y},\{f_{Y^n|X^n}\}^{\infty}_{i=1})$$The channel (without feedback) is called memoryless with a given marginal transition pdf $f_{Y|X}$ if $$f_{Y^n|X^n}(y^{n}|x^{n})=\prod_{i=1}^{n}f_{Y|X}(y_{i}|x_{i})$$for all $n\ge1$, $(x^{n},y^{n})\in(\mathcal{X}^{n}\times\mathcal{Y}^{n})$.