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The finite wiretap product law is stochastic

Statement

Use one-based coordinate labels for blocks: for $a^{n} \in A^{n}$ , a function on ${0, \dots, n - 1}$ , write $a_{i} : = a^{n} (i - 1)$ for $1 \leq i \leq n$ .

For finite nonempty alphabets $X, Y, Z$ , a stochastic kernel $W (y, z ∣ x)$ from $X$ to $Y \times Z$ , and every $n \geq 1$ , the block formula

$W^{n} (y^{n}, z^{n} ∣ x^{n}) : = \prod_{i = 1}^{n} W (y_{i}, z_{i} ∣ x_{i})$

is a stochastic kernel from $X^{n}$ to $(Y \times Z)^{n}$ .

Facts & Assumptions

Given: A finite stochastic kernel $W (y, z ∣ x)$ between the finite nonempty alphabets above and an integer $n \geq 1$ .

[F1]

Finite stochastic kernels: a finite stochastic kernel has nonnegative entries and each input row sums to one.

Proof

1.1givenF1

Fix $x^{n} \in X^{n}$ . Every factor $W (y_{i}, z_{i} ∣ x_{i})$ is nonnegative by [F1], so every product $W^{n} (y^{n}, z^{n} ∣ x^{n})$ is nonnegative.

2.1F1step 1.1algebra∎

Finite distributivity factors the output sum as $\sum_{y^{n}, z^{n}} W^{n} (y^{n}, z^{n} ∣ x^{n}) = \prod_{i = 1}^{n} (\sum_{y_{i}, z_{i}} W (y_{i}, z_{i} ∣ x_{i})) = \prod_{i = 1}^{n} 1 = 1.$ The row sums are one by [F1]; together with step 1.1 these are exactly the stochastic-kernel conditions.

Depends on

Finite stochastic kernels

Used by

Finite discrete memoryless wiretap channel and code model Definition
Strong-secrecy capacity of a finite DMC wiretap channel Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, arXiv:1001.3404v3 (standard reference, not scraped)