5

Here is a counterexample: Let $G_1$ be the digraph with vertex set $\mathbb N$, two loops at $0$, an edge from $0$ to $1$, and for every $i \geq 1$ an edge from $i$ to $(i+1)$ and two parallel edges from $i$ to $(i-1)$. Let $G_2$ be any countable digraph in which every vertex has $2$ outgoing edges and $4$ incoming edges, and let $f \colon V(G_2) \to V(G_1)...


2

The argument given in the book is actually quite misleading. It is much easier just to notice that the event $\{S_n/n\to \langle g,\pi\rangle \}$ from the definition of the function $g_\infty$ is (time shift) invariant, and therefore this function is harmonic by Theorem 17.1.3.


2

Studeny says in the abstract of his 1992 paper that: However, under the assumption that CIRs [conditional-independence relations] are grasped the existence of a countable characterization of CIRs is shown. Since Studeny seems to be calling a "complete finite axiomatization" a "finite characterization", this seems to suggest that Studeny is ...


1

This is the Ginibre ensemble, see Eigenvalue statistics of the real Ginibre ensemble for the eigenvalue distribution. For an $N\times N$ matrix with $N\gg 1$ there are on average $\sqrt{2N/\pi}$ eigenvalues on the real axis, uniformly in the interval $(-\sqrt N,\sqrt N$). The rest of the eigenvalues fill a disc of radius $\sqrt N$ in the complex plane, ...


1

Definitely no. Let $n = 2$ and $A := \{\omega \in \Omega \colon X_1(\omega) = X_2(\omega)\}$. Then $A \in \sigma(X_1,X_2)$, but (except in trivial cases) $A \not\in \sigma(X_1 + X_2)$.


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