Is the main part of certain exponential family sub-Gaussian?

Question

$X$ is in the form of exponential family i.e. $$\mathbb{P_\theta}x = h(x)e^{\langle \theta,T(x)\rangle-\phi(\theta)}$$ where $\theta\in \mathbb{R}^d$. If $\nabla\phi(\theta)$ is L-Lipschitz i.e. $$\vert\nabla\phi(\theta_1) - \nabla\phi(\theta_2)\vert \leq L\vert\theta_1-\theta_2\vert,$$ how can we prove $Z = \langle v, T(X)\rangle$ for fixed $v$ satisfying $\Vert v\Vert_2=1$ is sub-Gaussian?

I thought $Z$ should be like the linear combination of $X$, but I failed to reach what the Lipschitz condition can control here.

Answer 1

$\newcommand\th\theta\newcommand\la\lambda\newcommand\R{\mathbb R}$Note that for $t\in\R^d$ we have
$$M_\th(t):=E_\th e^{t\cdot T(X)} =\int_{\R^d}dx\,h(x)e^{(t+\th)\cdot T(x)-\phi(\th)} =e^{\phi(t+\th)-\phi(\th)},$$ where $\cdot$ denotes the dot product. So, $$E_\th T(X)=\nabla M_\th(0)=\nabla\phi(\th)$$ and, for any real $\la$, $$E_\th e^{\la(Z-E_\th Z)} =E_\th\exp\{\la v\cdot T(X)-\la v\cdot E_\th T(X))\} \\ =\exp\{\phi(\la v+\th)-\phi(\th)-\la v\cdot \nabla\phi(\th)\}. \tag{1}\label{1}$$ By the mean-value theorem, for some $a\in(0,1)$ depending on $\la,v,\th$, we have $$\phi(\la v+\th)-\phi(\th)=\la v\cdot\nabla\phi(a\la v+\th).$$ Also, by the Lipschitz condition on $\nabla\phi$, $$|\nabla\phi(a\la v+\th)-\nabla\phi(\th)|\le La\la|v|\le L\la.$$ Thus, $$\phi(\la v+\th)-\phi(\th)-\la v\cdot \nabla\phi(\th) =\la v\cdot(\nabla\phi(a\la v+\th)-\nabla\phi(\th)) \le L\la^2$$ and hence, by \eqref{1}, $$E_\th e^{\la(Z-E_\th Z)} \le e^{L\la^2},$$ so that $Z$ is (uniformly) sub-Gaussian for all $\th$.