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\title{A Probabilistic Proof of a Positivstellensatz of Baldi-Sinn-Telek-Weigert}
\author{}
\date{17 Oct 2026}

\begin{document}
\maketitle

\begin{abstract}
The AI gives a probabilistic proof of a uniform-denominator Positivstellensatz of Baldi--Sinn--Telek--Weigert for Laurent polynomials whose induced homogeneous form is strictly positive on the nonnegative part of a projective toric variety. Only this abstract was human-written. Three classical tools from mathematical statistics and probability theory were used by the AI: the Barndorff-Nielsen completion of discrete exponential families, sufficiency-based invariance under exponential tilting following Halmos--Savage and Van Campenhout--Cover, and the conditional limit theorem (equivalence of ensembles) of Diaconis--Freedman.
\end{abstract}

\section{Exponential Families on Lattice Configurations and Toric Closures}

Let $\mathcal{A} = \{a_1, \dots, a_m\} \subset \Z^n$ be a non-empty finite set of lattice points, and let $P := \conv(\mathcal{A}) \subset \R^n$ denote its convex support. For any non-empty face $F \preceq P$, we write $\mathcal{A}_F := \mathcal{A} \cap F$, so that $F = \conv(\mathcal{A}_F)$.

The general theory of natural exponential families generated by a carrier measure on $\R^n$ was systematically developed by Barndorff-Nielsen~\cite[Chapter~8]{BarndorffNielsen1978} and Brown~\cite[Chapter~1]{Brown1986}, and is surveyed from the perspective of graphical models and variational inference by Wainwright--Jordan~\cite[Section~3]{WainwrightJordan2008}. We recall the standard definition following Brown~\cite[Chapter~1]{Brown1986}.

\begin{definition}[{Brown~\cite[Chapter~1]{Brown1986}}]\label{def:exp_family}
Let $\nu$ be the counting measure on the finite set $\mathcal{A} \subset \Z^n$. The \emph{natural exponential family} $\mathcal{E}_{\mathcal{A}} = \{P_\theta : \theta \in \R^n\}$ generated by $\nu$ with natural parameter $\theta \in \R^n$ is the family of probability distributions on $\mathcal{A}$ with probability mass function
\begin{equation}\label{eq:Gibbs}
P_\theta(X = a) := \exp\bigl(\langle \theta, a \rangle - \Lambda_{\mathcal{A}}(\theta)\bigr), \qquad a \in \mathcal{A},
\end{equation}
where $\Lambda_{\mathcal{A}}(\theta) := \log \sum_{a' \in \mathcal{A}} e^{\langle \theta, a' \rangle}$ is the \emph{cumulant generating function} (or \emph{log-partition function}). Under the multiplicative reparameterization $t = e^\theta := (e^{\theta_1}, \dots, e^{\theta_n}) \in \R_{>0}^n$, we have $P_\theta(X = a) = t^a / \sum_{a' \in \mathcal{A}} t^{a'}$.
\end{definition}

Because $\mathcal{A}$ is finite, the natural parameter space is $\Theta = \R^n$, so $\mathcal{E}_{\mathcal{A}}$ is a full, regular exponential family in the sense of Brown~\cite[Chapter~1]{Brown1986}. By Brown~\cite[Theorem~2.2]{Brown1986}, $\Lambda_{\mathcal{A}}$ is real-analytic and convex on $\R^n$, and its gradient and Hessian are the mean vector and covariance matrix of $X \sim P_\theta$:
\begin{align}
\mu_{\mathcal{A}}(\theta) &:= \nabla \Lambda_{\mathcal{A}}(\theta) = \E_{P_\theta}[X] = \sum_{a \in \mathcal{A}} a \, P_\theta(X = a), \notag\\ \Sigma_{\mathcal{A}}(\theta) &:= \nabla^2 \Lambda_{\mathcal{A}}(\theta) = \Cov_{P_\theta}(X). \label{eq:moments}
\end{align}
For each non-empty face $F \preceq P$, let $V_F := \spanR(\mathcal{A}_F - a_0)$ (for any fixed $a_0 \in \mathcal{A}_F$) be the linear subspace parallel to the affine hull of $F$, of dimension $d_F := \dim(F)$. By Brown~\cite[Proposition~1.4]{Brown1986}, the covariance operator $\Sigma_{\mathcal{A}_F}(\theta)$ of the facial exponential family $\mathcal{E}_{\mathcal{A}_F}$ supported on $\mathcal{A}_F$ restricts to a strictly positive definite quadratic form on $V_F$ for every $\theta \in \R^n$.

A foundational result in the theory of exponential families with finite support describes the topological closure $\overline{\mathcal{E}}_{\mathcal{A}}$ of $\mathcal{E}_{\mathcal{A}}$ inside the probability simplex $\Delta_{m-1} = \{p \in \R_{\ge 0}^{\mathcal{A}} : \sum_{a \in \mathcal{A}} p_a = 1\}$ together with the behavior of the mean-value map $\mu : Q \mapsto \E_Q[X]$. The homeomorphism between the parameter space of a log-linear model and the interior of its marginal polytope originates in the classical theorem of Birch~\cite{Birch1963} and was established for general regular exponential families by Barndorff-Nielsen~\cite[Theorem~9.1]{BarndorffNielsen1978} and Brown~\cite[Theorem~3.6]{Brown1986}. The facial stratification of the boundary completion $\overline{\mathcal{E}}_{\mathcal{A}}$ was proved for finite support by Barndorff-Nielsen~\cite[Theorem~9.2]{BarndorffNielsen1978} (where $\overline{\mathcal{E}}_{\mathcal{A}}$ is termed the \emph{aggregate exponential family}; see also Brown~\cite[Chapter~6]{Brown1986}, Csisz\'ar--Mat\'u\v{s}~\cite{CsiszarMatus2005}, Geyer~\cite[Theorem~4]{Geyer2009}, and Rauh--Kahle--Ay~\cite{RauhKahleAy2011}). In algebraic statistics, Geiger--Meek--Sturmfels~\cite[Theorem~3.2]{GeigerMeekSturmfels2006} showed that $\overline{\mathcal{E}}_{\mathcal{A}}$ coincides with the nonnegative part $(X_{\mathcal{A}})_{\ge 0} := X_{\mathcal{A}} \cap \Delta_{m-1}$ of the projective toric variety $X_{\mathcal{A}} \subset \mathbb{P}^{m-1}$, under which the mean-value map $\mu$ is the algebraic moment map. We state the completion theorem following the formulation of Barndorff-Nielsen~\cite[Theorem~9.2]{BarndorffNielsen1978}.

\begin{theorem}[{Barndorff-Nielsen~\cite[Theorem~9.2]{BarndorffNielsen1978}}]\label{thm:BN_completion}
Let $\mathcal{E}_{\mathcal{A}}$ be the natural exponential family generated by the counting measure on the finite set $\mathcal{A} \subset \Z^n$ with convex support $P = \conv(\mathcal{A})$, and for each non-empty face $F \preceq P$, let $\mathcal{E}_{\mathcal{A}_F}$ be the natural exponential family generated by the counting measure on $\mathcal{A}_F = \mathcal{A} \cap F$. Define the aggregate exponential family
\begin{equation}\label{eq:stratification}
\overline{\mathcal{E}}_{\mathcal{A}} := \bigsqcup_{F \preceq P} \mathcal{E}_{\mathcal{A}_F} \subset \Delta_{m-1}.
\end{equation}
Then:
\begin{enumerate}[label=(\roman*)]
\item $\overline{\mathcal{E}}_{\mathcal{A}}$ is the closure of $\mathcal{E}_{\mathcal{A}}$ in the space of probability measures on $\mathcal{A}$ under the weak (equivalently, total variation) topology, and hence is compact.
\item For each non-empty face $F \preceq P$, the mean-value map $\mu_F : \mathcal{E}_{\mathcal{A}_F} \to \relint(F)$ given by $P_{F,\theta} \mapsto \E_{P_{F,\theta}}[X]$ is a homeomorphism (and a real-analytic diffeomorphism when parameterized by $\theta \in V_F$).
\item The extended mean-value map
\begin{equation}\label{eq:moment_map}
\mu : \overline{\mathcal{E}}_{\mathcal{A}} \longrightarrow P = \conv(\mathcal{A}), \qquad Q \longmapsto \E_Q[X] = \sum_{a \in \mathcal{A}} Q(a)\, a,
\end{equation}
is a bijection and a homeomorphism of compact topological spaces.
\end{enumerate}
\end{theorem}

\section{Probabilistic Representation of $\mathcal{A}$-Copositivity and Toric Multipliers}

For any integer $m \ge 1$ and $u \in \Z^n$, we denote by $m\cdot\mathcal{A} := \{a^{(1)} + \dots + a^{(m)} : a^{(1)}, \dots, a^{(m)} \in \mathcal{A}\}$ the $m$-fold Minkowski sum of $\mathcal{A}$, and by
\begin{equation}\label{eq:walk_count}
W_m(u) := [t^u]\!\left(\sum_{a \in \mathcal{A}} t^a\right)^{\!m} = \#\bigl\{(a^{(1)}, \dots, a^{(m)}) \in \mathcal{A}^m : a^{(1)} + \dots + a^{(m)} = u\bigr\}
\end{equation}
the number of $m$-step lattice walks with steps in $\mathcal{A}$ ending at $u$. Clearly $W_m(u) > 0$ if and only if $u \in m\cdot\mathcal{A}$. Similarly, for each non-empty face $F \preceq P$, we write $W_{m,F}(u) := \#\{(a^{(1)}, \dots, a^{(m)}) \in \mathcal{A}_F^m : a^{(1)} + \dots + a^{(m)} = u\}$.

\begin{lemma}[Facial Walk Restriction]\label{lem:facial_walk}
Let $F \preceq P$ be a non-empty face and let $m \ge 1$. Then $(m\cdot\mathcal{A}) \cap mF = m\cdot\mathcal{A}_F$, and for every $u \in m\cdot\mathcal{A}_F$,
\begin{equation}
W_m(u) = W_{m,F}(u).
\end{equation}
\end{lemma}

\begin{proof}
Since $F$ is a face of $P = \conv(\mathcal{A})$, there exist an inward-pointing normal vector $v_F \in \R^n$ and a scalar $\lambda_F \in \R$ such that $\langle v_F, a \rangle = \lambda_F$ for all $a \in \mathcal{A}_F$ and $\langle v_F, a \rangle > \lambda_F$ for all $a \in \mathcal{A} \setminus \mathcal{A}_F$. For any $m$-tuple $(a^{(1)}, \dots, a^{(m)}) \in \mathcal{A}^m$ with sum $u = \sum_{j=1}^m a^{(j)}$, we have
\[
\langle v_F, u \rangle = \sum_{j=1}^m \langle v_F, a^{(j)} \rangle \ge m\lambda_F,
\]
with equality (which holds if and only if $u \in mF$) if and only if $\langle v_F, a^{(j)} \rangle = \lambda_F$ for every $j = 1, \dots, m$, i.e., $a^{(j)} \in \mathcal{A}_F$ for all $j$. Thus every $m$-step walk in $\mathcal{A}$ summing to $u \in mF$ uses solely steps in $\mathcal{A}_F$.
\end{proof}

Fix $k \in \N$ and let $f(t) = \sum_{b \in k\cdot\mathcal{A}} c_b t^b \in \R[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$ be a Laurent polynomial whose support is contained in $k\cdot\mathcal{A}$. For each face $F \preceq P$, let $f_F(t) := \sum_{b \in k\cdot\mathcal{A}_F} c_b t^b$ denote the facial truncation of $f$ to $kF$ (analogous to the facial sub-polynomials on the simplex studied by Castle--Powers--Reznick~\cite{CastlePowersReznick2011} and Dickinson--Povh~\cite{DickinsonPovh2014}). Following Baldi--Sinn--Telek--Weigert~\cite[Section~2.2]{BaldiSinnTelekWeigert2025}, $f$ is \emph{strictly $\mathcal{A}$-copositive} if the homogeneous form induced by $f$ is strictly positive on the nonnegative projective toric variety $(X_{\mathcal{A}})_{\ge 0}$. By the facial characterization in Baldi--Sinn--Telek--Weigert~\cite[Proposition~2.7]{BaldiSinnTelekWeigert2025}, $f$ is strictly $\mathcal{A}$-copositive if and only if $f_F(t) > 0$ for every non-empty face $F \preceq P$ and every $t \in \R_{>0}^n$.

Define the normalized coefficient observable $g : k\cdot\mathcal{A} \to \R$ by
\begin{equation}\label{eq:observable}
g(b) := \frac{c_b}{W_k(b)}, \qquad b \in k\cdot\mathcal{A}.
\end{equation}

\begin{lemma}[Unconditional Gibbs Expectation and Uniform Positivity]\label{lem:unconditional}
For any probability measure $Q \in \overline{\mathcal{E}}_{\mathcal{A}}$, let $X_1, \dots, X_k$ be independent and identically distributed (i.i.d.) random vectors with law $Q$, and set $S_k := X_1 + \dots + X_k \in k\cdot\mathcal{A}$.
\begin{enumerate}[label=(\roman*)]
\item If $Q = P_{F,\theta} \in \mathcal{E}_{\mathcal{A}_F}$ for a face $F \preceq P$ and $\theta \in \R^n$ (with $t = e^\theta$), then
\begin{equation}\label{eq:uncond_id}
\E_{P_{F,\theta}}[g(S_k)] = \frac{f_F(t)}{\left(\sum_{a \in \mathcal{A}_F} t^a\right)^{\!k}}.
\end{equation}
\item The Laurent polynomial $f$ is strictly $\mathcal{A}$-copositive if and only if there exists a constant $\delta > 0$ such that
\begin{equation}\label{eq:uniform_delta}
\inf_{Q \in \overline{\mathcal{E}}_{\mathcal{A}}} \E_Q[g(S_k)] = \delta > 0.
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Under $P_{F,\theta}^{\otimes k}$, any $k$-tuple $(a^{(1)}, \dots, a^{(k)}) \in \mathcal{A}_F^k$ with sum $b \in k\cdot\mathcal{A}_F$ has probability $t^b / (\sum_{a \in \mathcal{A}_F} t^a)^k$. Summing over all $W_{k,F}(b)$ such $k$-tuples gives $\Prob_{P_{F,\theta}}(S_k = b) = W_{k,F}(b) t^b / (\sum_{a \in \mathcal{A}_F} t^a)^k$. By Lemma~\ref{lem:facial_walk}, $W_{k,F}(b) = W_k(b)$ for all $b \in k\cdot\mathcal{A}_F$, whence
\[
\E_{P_{F,\theta}}[g(S_k)] = \sum_{b \in k\cdot\mathcal{A}_F} \frac{c_b}{W_k(b)} \frac{W_{k,F}(b) t^b}{\left(\sum_{a \in \mathcal{A}_F} t^a\right)^{\!k}} = \frac{\sum_{b \in k\cdot\mathcal{A}_F} c_b t^b}{\left(\sum_{a \in \mathcal{A}_F} t^a\right)^{\!k}} = \frac{f_F(t)}{\left(\sum_{a \in \mathcal{A}_F} t^a\right)^{\!k}}.
\]
(ii) Since $\mathcal{A}$ is finite, the map $Q \mapsto \E_Q[g(S_k)] = \sum_{(a^{(1)},\dots,a^{(k)}) \in \mathcal{A}^k} g(a^{(1)} + \dots + a^{(k)}) \prod_{j=1}^k Q(a^{(j)})$ is a polynomial in the probabilities $(Q(a))_{a \in \mathcal{A}}$ and is therefore continuous on $\overline{\mathcal{E}}_{\mathcal{A}}$. By Theorem~\ref{thm:BN_completion}(i) and part~(i), $f_F(t) > 0$ for all $F \preceq P$ and $t \in \R_{>0}^n$ if and only if $\E_Q[g(S_k)] > 0$ for all $Q \in \overline{\mathcal{E}}_{\mathcal{A}}$, which by the compactness of $\overline{\mathcal{E}}_{\mathcal{A}}$ in Theorem~\ref{thm:BN_completion}(i) is equivalent to \eqref{eq:uniform_delta}.
\end{proof}

We now connect the coefficients of the toric multiplier $(\sum_{a \in \mathcal{A}} t^a)^N f(t)$ to conditioning on a sufficient statistic. By the classical factorization criterion for sufficient statistics due to Halmos--Savage~\cite[Theorem~1]{HalmosSavage1949} (see also Lehmann--Romano~\cite[Theorem~2.7.1]{LehmannRomano2005}), the sample sum $S_{N+k} = \sum_{i=1}^{N+k} X_i$ is a minimal sufficient statistic for the natural parameter $\theta$ of the exponential family $\mathcal{E}_{\mathcal{A}}$. As exploited in the exponential tilting method of Esscher~\cite{Esscher1932}, Van Campenhout--Cover~\cite[Equation~(9)]{VanCampenhoutCover1981}, and Diaconis--Freedman~\cite[Section~2]{DiaconisFreedman1988}, conditioning an i.i.d.\ sample from $P_\theta$ on $S_{N+k} = \gamma$ eliminates $\theta$ and reduces to the uniform microcanonical distribution over lattice trajectories summing to $\gamma$.

\begin{lemma}[Microcanonical Conditional Representation of Toric Multipliers]\label{lem:sufficiency}
Let $N \in \N$, and let $X_1, \dots, X_{N+k}$ be i.i.d.\ random vectors taking values in $\mathcal{A}$ with law $P_\theta \in \mathcal{E}_{\mathcal{A}}$. Set $S_m := X_1 + \dots + X_m$ for $1 \le m \le N+k$.
\begin{enumerate}[label=(\roman*)]
\item For every $\gamma \in (N+k)\cdot\mathcal{A}$, the conditional distribution of $(X_1, \dots, X_{N+k})$ given $S_{N+k} = \gamma$ is independent of $\theta \in \R^n$ and is the uniform discrete distribution on the set of trajectories $\{(x_1, \dots, x_{N+k}) \in \mathcal{A}^{N+k} : \sum_{i=1}^{N+k} x_i = \gamma\}$.
\item For every $\gamma \in (N+k)\cdot\mathcal{A}$ and $b \in k\cdot\mathcal{A}$, the conditional distribution of $S_k$ given $S_{N+k} = \gamma$ satisfies
\begin{align}
\Prob(S_k = b \mid S_{N+k} = \gamma) &= \frac{W_k(b) W_N(\gamma - b)}{W_{N+k}(\gamma)} \notag \\
&= \Prob_{P_\theta}(S_k = b) \frac{\Prob_{P_\theta}(S_{N+k} - S_k = \gamma - b)}{\Prob_{P_\theta}(S_{N+k} = \gamma)}, \label{eq:cond_law}
\end{align}
and the coefficient of $t^\gamma$ in $(\sum_{a \in \mathcal{A}} t^a)^N f(t)$ satisfies the exact identity
\begin{equation}\label{eq:coeff_cond_exp}
\frac{[t^\gamma]\!\left(\left(\sum_{a \in \mathcal{A}} t^a\right)^{\!N} f(t)\right)}{W_{N+k}(\gamma)} = \E\bigl[g(S_k) \;\big|\; S_{N+k} = \gamma\bigr].
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
Under $P_\theta^{\otimes (N+k)}$, the joint probability mass function of $(X_1, \dots, X_{N+k})$ at $(x_1, \dots, x_{N+k}) \in \mathcal{A}^{N+k}$ is
\[
\Prob_{P_\theta}(X_1 = x_1, \dots, X_{N+k} = x_{N+k}) = \exp\Bigl(\Bigl\langle \theta, \sum_{i=1}^{N+k} x_i \Bigr\rangle - (N+k)\Lambda_{\mathcal{A}}(\theta)\Bigr),
\]
which depends on $(x_1, \dots, x_{N+k})$ only through $\sum_{i=1}^{N+k} x_i$. Thus all $W_{N+k}(\gamma)$ trajectories in $\mathcal{A}^{N+k}$ summing to $\gamma$ have equal conditional probability $1/W_{N+k}(\gamma)$ regardless of $\theta$, proving (i). Partitioning the $N+k$ steps into the first $k$ steps summing to $b$ and the last $N$ steps summing to $\gamma - b$ gives the first equality in \eqref{eq:cond_law}, while evaluating the conditional probability under $P_\theta^{\otimes (N+k)}$ gives the second equality. Finally, expanding $(\sum_{a \in \mathcal{A}} t^a)^N f(t) = (\sum_{u \in N\cdot\mathcal{A}} W_N(u) t^u)(\sum_{b \in k\cdot\mathcal{A}} c_b t^b)$ yields
\begin{align}
\frac{[t^\gamma]\!\left(\left(\sum_{a \in \mathcal{A}} t^a\right)^{\!N} f(t)\right)}{W_{N+k}(\gamma)} &= \sum_{b \in k\cdot\mathcal{A}} \frac{c_b}{W_k(b)} \frac{W_k(b) W_N(\gamma - b)}{W_{N+k}(\gamma)} \notag\\
&= \E\bigl[g(S_k) \;\big|\; S_{N+k} = \gamma\bigr]. \qedhere
\end{align}
\end{proof}

\section{Equivalence of Ensembles and the Proof of Theorem~A}

The asymptotic convergence of the conditional distribution of the first $k$ coordinates $(X_1, \dots, X_k)$ given the sufficient statistic $S_M = \gamma'$ to the i.i.d.\ canonical exponential family distribution $P_{\theta(\gamma'/M)}^{\otimes k}$ is known in statistical mechanics and information theory as the \emph{equivalence of ensembles} or the \emph{Gibbs conditioning principle}. Pointwise versions of this principle were proved by Martin-L\"of~\cite{MartinLof1979}, Van Campenhout--Cover~\cite[Theorem~1]{VanCampenhoutCover1981}, Csisz\'ar~\cite{Csiszar1984}, and Zabell~\cite{Zabell1993} by combining exponential tilting with the multivariate local central limit theorem for lattice random vectors established by Rvaceva~\cite{Rvaceva1962}, Petrov~\cite{Petrov1975}, Spitzer~\cite[Proposition~P9, p.~75]{Spitzer1976}, and Bhattacharya--Ranga Rao~\cite[Theorem~22.1]{BhattacharyaRangaRao1986}. A quantitative version holding uniformly over compact subsets of the interior of the mean space in total variation distance $\left\|\mu - \nu\right\|_{\TV} := \sup_{E} |\mu(E) - \nu(E)|$ was established by Diaconis--Freedman~\cite[Theorem~1.1 and Proposition~2.6]{DiaconisFreedman1988}. We state the theorem of Diaconis--Freedman~\cite[Theorem~1.1]{DiaconisFreedman1988} specialized to the facial exponential family $\mathcal{E}_{\mathcal{A}_F}$.

\begin{theorem}[{Diaconis--Freedman~\cite[Theorem~1.1]{DiaconisFreedman1988}}]\label{thm:DF_conditional}
Let $F \preceq P$ be a non-empty face, and let $K \subset \relint(F)$ be a compact subset. For integers $1 \le k \le M$ and $\gamma' \in M\cdot\mathcal{A}_F$, let $Q_{F, \gamma', k}$ denote the conditional distribution of $(X_1, \dots, X_k)$ given $S_M = \gamma'$ under the exponential family $\mathcal{E}_{\mathcal{A}_F}$. Then there exists a constant $C_K < \infty$, depending only on $\mathcal{A}_F$ and $K$, such that for all $M \ge k$ and all $\gamma' \in M\cdot\mathcal{A}_F$ satisfying $\gamma'/M \in K$,
\begin{equation}\label{eq:DF_TV}
\left\|Q_{F, \gamma', k} - P_{F, \theta(\gamma'/M)}^{\otimes k}\right\|_{\TV} \le C_K \frac{k}{M},
\end{equation}
where $\theta(\gamma'/M) := \mu_F^{-1}(\gamma'/M) \in V_F$ is the unique parameter in $V_F$ satisfying $\E_{P_{F,\theta(\gamma'/M)}}[X_1] = \gamma'/M$.
\end{theorem}

We now prove the toric extension of the theorem of P\'olya~\cite{Polya1928} and the Laurent polyhedral theorem of Handelman~\cite{Handelman1988}, stated as Theorem~A in the introduction and as Theorem~3.6 in the text of Baldi--Sinn--Telek--Weigert~\cite{BaldiSinnTelekWeigert2025}.

\begin{theorem}[{Baldi--Sinn--Telek--Weigert~\cite[Theorem~3.6]{BaldiSinnTelekWeigert2025}}]\label{thm:main}
Let $\mathcal{A} = \{a_1, \dots, a_m\} \subset \Z^n$ be a finite set and $k \in \N$. Let $f = \sum_{b \in k\cdot\mathcal{A}} c_b t^b$ be a Laurent polynomial with support contained in $k\cdot\mathcal{A}$. Then $f$ is strictly $\mathcal{A}$-copositive if and only if there exists $N \in \N$ such that $\left(\sum_{a \in \mathcal{A}} t^a\right)^N f$ has only nonnegative coefficients and $\Newt\!\left(\left(\sum_{a \in \mathcal{A}} t^a\right)^N f\right) = \conv((N+k)\cdot\mathcal{A})$. Moreover, if $f$ is strictly $\mathcal{A}$-copositive, then for $N \in \N$ large enough, every monomial with exponent in $(N+k)\cdot\mathcal{A}$ appears in $\left(\sum_{a \in \mathcal{A}} t^a\right)^N f$ with strictly positive coefficient.
\end{theorem}

\begin{proof}
The ``if'' direction is immediate: if $h_N(t) := (\sum_{a \in \mathcal{A}} t^a)^N f(t)$ has nonnegative coefficients and $\Newt(h_N) = \conv((N+k)\cdot\mathcal{A}) = (N+k)P$, then for every non-empty face $F \preceq P$, the facial truncation $(h_N)_F(t) = (\sum_{a \in \mathcal{A}_F} t^a)^N f_F(t)$ is a non-zero Laurent polynomial with nonnegative coefficients, so $f_F(t) > 0$ for all $t \in \R_{>0}^n$.

For the ``only if'' direction and the ``moreover'' statement, assume that $f$ is strictly $\mathcal{A}$-copositive. By Lemma~\ref{lem:unconditional}(ii), $\delta := \inf_{Q \in \overline{\mathcal{E}}_{\mathcal{A}}} \E_Q[g(S_k)] > 0$. By Lemma~\ref{lem:sufficiency}(ii), to prove that every monomial with exponent $\gamma \in (N+k)\cdot\mathcal{A}$ appears in $(\sum_{a \in \mathcal{A}} t^a)^N f(t)$ with strictly positive coefficient for all sufficiently large $N$, it suffices to show that there exists $N_0 \in \N$ such that
\begin{equation}\label{eq:goal}
\min_{\gamma \in (N+k)\cdot\mathcal{A}} \E\bigl[g(S_k) \;\big|\; S_{N+k} = \gamma\bigr] \ge \frac{\delta}{2} > 0 \qquad \text{for all } N \ge N_0.
\end{equation}
Suppose, for the sake of contradiction, that \eqref{eq:goal} fails. Then there exist a strictly increasing sequence of positive integers $N_j \to \infty$ and a sequence of lattice points $\gamma_j \in (N_j + k)\cdot\mathcal{A}$ such that
\begin{equation}\label{eq:contra}
\E\bigl[g(S_k) \;\big|\; S_{N_j+k} = \gamma_j\bigr] < \frac{\delta}{2} \qquad \text{for all } j \ge 1.
\end{equation}
Since $\mu_j := \frac{\gamma_j}{N_j + k} \in P = \conv(\mathcal{A})$ and $P$ is compact, after passing to a subsequence we may assume that
\[
\lim_{j \to \infty} \mu_j = \mu_\infty \in P.
\]
Since $P = \bigsqcup_{F \preceq P} \relint(F)$, there is a unique non-empty face $F \preceq P$ such that $\mu_\infty \in \relint(F)$.

Choose an inward-pointing normal vector $v_F \in \R^n$ and scalar $\lambda_F \in \R$ for the face $F$ as in Lemma~\ref{lem:facial_walk}, and define the nonnegative facial discrepancy $\psi_F(a) := \langle v_F, a \rangle - \lambda_F$ for $a \in \mathcal{A}$. Then $\psi_F(a) = 0$ for all $a \in \mathcal{A}_F$, while if $\mathcal{A} \setminus \mathcal{A}_F \neq \emptyset$, then $\eta_F := \min_{a \in \mathcal{A} \setminus \mathcal{A}_F} \psi_F(a) > 0$. (If $\mathcal{A} \setminus \mathcal{A}_F = \emptyset$, then $F = P$ and $M_j = 0$ below.)

For each $j \ge 1$, consider the random walk $(X_1, \dots, X_{N_j+k}) \in \mathcal{A}^{N_j+k}$ conditioned on $S_{N_j+k} = \gamma_j$. Let $I_{\mathrm{off}} := \{i \in \{1, \dots, N_j+k\} : X_i \notin \mathcal{A}_F\}$ be the random set of indices of off-face steps, let $M_j := |I_{\mathrm{off}}|$ be its cardinality, and let $U_j := \sum_{i \in I_{\mathrm{off}}} X_i$ be the sum of the off-face steps. On the conditioning event $\{S_{N_j+k} = \gamma_j\}$, we have the deterministic inequality
\[
\eta_F M_j \le \sum_{i=1}^{N_j+k} \psi_F(X_i) = \langle v_F, \gamma_j \rangle - (N_j + k)\lambda_F = (N_j + k)\bigl(\langle v_F, \mu_j \rangle - \lambda_F\bigr).
\]
Since $\mu_j \to \mu_\infty \in F$ and $\langle v_F, \mu_\infty \rangle = \lambda_F$, the deterministic upper bound
\begin{equation}\label{eq:m_max}
m_j^{\max} := \frac{N_j + k}{\eta_F}\bigl(\langle v_F, \mu_j \rangle - \lambda_F\bigr) = o(N_j) \qquad (j \to \infty)
\end{equation}
satisfies $0 \le M_j \le m_j^{\max}$ and $\|U_j\| \le m_j^{\max} \max_{a \in \mathcal{A}} \|a\| = o(N_j)$ for \emph{every} trajectory in $\{S_{N_j+k} = \gamma_j\}$.

Let $E_j := \{I_{\mathrm{off}} \cap \{1, \dots, k\} = \emptyset\}$ be the event that the first $k$ steps $X_1, \dots, X_k$ all lie in $\mathcal{A}_F$. Because the uniform conditional distribution on $\{S_{N_j+k} = \gamma_j\}$ is exchangeable (invariant under permutations of the $N_j+k$ coordinates, as noted in Diaconis--Freedman~\cite[Section~1]{DiaconisFreedman1988}), the conditional probability of $E_j^c$ satisfies
\begin{align}
\Prob\bigl(E_j^c \;\big|\; S_{N_j+k} = \gamma_j\bigr) &\le \sum_{i=1}^k \Prob\bigl(X_i \notin \mathcal{A}_F \;\big|\; S_{N_j+k} = \gamma_j\bigr) \notag\\
&= \frac{k\, \E[M_j \mid S_{N_j+k} = \gamma_j]}{N_j + k} \le \frac{k\, m_j^{\max}}{N_j + k} \xrightarrow{j \to \infty} 0. \label{eq:Ej_bound}
\end{align}

On the event $E_j$, further condition on the pair $(I_{\mathrm{off}}, U_j)$. Given $(I_{\mathrm{off}}, U_j)$ with $I_{\mathrm{off}} \subset \{k+1, \dots, N_j+k\}$ and $|I_{\mathrm{off}}| = m \le m_j^{\max}$, the remaining $M_j' := N_j + k - m$ random variables $(X_i)_{i \notin I_{\mathrm{off}}}$ (which include $X_1, \dots, X_k$) are conditionally a uniform random walk of length $M_j'$ on $\mathcal{A}_F$ conditioned on their sum being
\[
\gamma_j' := \gamma_j - U_j \in M_j'\cdot\mathcal{A}_F,
\]
so the conditional law of $(X_1, \dots, X_k)$ given $(S_{N_j+k} = \gamma_j, E_j, I_{\mathrm{off}}, U_j)$ is precisely $Q_{F, \gamma_j', k}$. Moreover, uniformly over all possible realizations of $(I_{\mathrm{off}}, U_j)$ on $\{S_{N_j+k} = \gamma_j\}$, the empirical mean $\mu_j' := \gamma_j' / M_j'$ satisfies
\[
\mu_j' = \frac{\gamma_j - U_j}{N_j + k - m} = \frac{\mu_j - U_j/(N_j+k)}{1 - m/(N_j+k)} \xrightarrow{j \to \infty} \mu_\infty \in \relint(F).
\]
Consequently, fixing a compact neighborhood $K \subset \relint(F)$ of $\mu_\infty$, we have $\mu_j' \in K$ for every realization of $(I_{\mathrm{off}}, U_j)$ whenever $j$ is sufficiently large. Applying Theorem~\ref{thm:DF_conditional} of Diaconis--Freedman~\cite[Theorem~1.1]{DiaconisFreedman1988} on $\mathcal{A}_F$, we obtain, uniformly over all realizations of $(I_{\mathrm{off}}, U_j)$ on $E_j$,
\begin{align}
&\left|\E\bigl[g(S_k) \;\big|\; S_{N_j+k} = \gamma_j, \, E_j, \, I_{\mathrm{off}}, \, U_j\bigr] - \E_{P_{F,\theta(\mu_j')}}[g(S_k)]\right| \notag\\
\le {} & 2\|g\|_\infty \left\|Q_{F, \gamma_j', k} - P_{F, \theta(\mu_j')}^{\otimes k}\right\|_{\TV} \\
\le {} & \frac{2\|g\|_\infty C_K k}{N_j + k - m_j^{\max}} \xrightarrow{j \to \infty} 0,
\end{align}
where $\|g\|_\infty := \max_{b \in k\cdot\mathcal{A}} |g(b)| < \infty$. By the continuity of $\mu' \mapsto \theta(\mu') = \mu_F^{-1}(\mu')$ in Theorem~\ref{thm:BN_completion}(ii), $\E_{P_{F,\theta(\mu_j')}}[g(S_k)] \to \E_{P_{F,\theta(\mu_\infty)}}[g(S_k)]$ uniformly as $j \to \infty$. Combining this with $\Prob(E_j^c \mid S_{N_j+k} = \gamma_j) \to 0$ from \eqref{eq:Ej_bound} and the law of total expectation yields
\[
\lim_{j \to \infty} \E\bigl[g(S_k) \;\big|\; S_{N_j+k} = \gamma_j\bigr] = \E_{P_{F,\theta(\mu_\infty)}}[g(S_k)] \ge \delta,
\]
which contradicts \eqref{eq:contra}. Finally, since every coefficient of $(\sum_{a \in \mathcal{A}} t^a)^N f(t)$ on $(N+k)\cdot\mathcal{A}$ is strictly positive for $N \ge N_0$, its support equals $(N+k)\cdot\mathcal{A}$ and its Newton polytope equals $\conv((N+k)\cdot\mathcal{A})$.
\end{proof}

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\end{document}