Some questions about the number of moments for random variables

Question

I'm reading a paper about econometrics and meeting some questions, and I'm not sure I can describe them perfect. $s_t$ is a scaled score function, and $f_t$ is a stochastic time-varying parameter, and by calculation, we can get \begin{align} p_y(y_t\vert f_t;\lambda) &=\frac{\Gamma\left(\frac{\lambda+1}{2}\right)}{\Gamma\left(\frac{\lambda}{2}\right)\sqrt{\pi\lambda f_t}} \left(1+\frac{y_t^2}{\lambda f_t}\right)^{-\frac{\lambda+1}{2}}\\ s_t &= \frac{(\lambda+1)y_t^2}{\lambda+\frac{y_t^2}{f_t}}-f_t \end{align} Now we can get $$\vert s_t\vert\leq\sup_{y_t}\vert s_t\vert<c_1\cdot\vert f_t\vert$$ for some constant $c_1$. My problem is, how to get the result that the number of moments of $\sup_{\theta}s_t$ is smaller than the number of moments of $f_t$, where $\theta$ is a parameter vector. And I wanna know where can I find something about number of moments because there are some other problems about this. Thanks very much.