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## Hot answers tagged pr.probability

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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