$X$ 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.