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Consider the space $$C^0(\mathbb{Z}^n)= \{f:\mathbb{Z}^n \to \mathbb{R} \,\,|\,\, f\text{ is bounded} \}.$$ Suppose we have a probability distribution $p$ on $\mathbb{Z}^n$ with mean zero: \begin{align*}p:\mathbb{Z}^n &\to \mathbb{R} \\\\ p(\mathbf{n})&\geq 0 \quad \text{for all}\quad \mathbf{n}\in \mathbb{Z}^n\\\\ \sum_{\mathbf{n}\in \mathbb{Z}^n} p(\mathbf{n})&=1 \\\\\sum_{\mathbf{n}\in \mathbb{Z}^n} … Continue reading →