Learning the distribution of a continuous variable using LSTM

I am trying to implement the following paper : https://arxiv.org/pdf/2006.10701.pdf. In order, to estimate the priors of the hidden states which have continuous values, the authors use a LSTM. I have never done this before, and I was wondering If:

• It is possible to directly do this, or should I make an assumption about the distribution and use the reparameterization trick, or is estimating p(z_{i}|z_{i-1}) equivalent to estimating z_{i} ?
• In the HalfCheetah environment, the z_{i} is a function of the episod. When trying to estimate p(z_{i}|z_{i-1}) doesn’t the way of feeding data affect the learned distribution ?