Properties of Student t statistic

For $n=1, 2, 3, \dots$, let the joint moment generating function of $(X, Y_n)$ be $$M_{X, Y_n}(t_1, t_2)=e^{\frac{t_1^2}{2}}(1-2t_2)^{-\frac{n}{2}},\quad t_1\in\mathbb{R}, t_2 < \frac{1}{2}.$$ If $T_n = \frac{\sqrt{n}X}{\sqrt{Y_n}}, n\ge 1$, then which one of the following statements is true?