Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq k\leq i$$ for all $i\geq 1$, where $p_i$ is the $i$th prime, e.g. $p_1=2,\ p_2=3$ etc. Does the entropy rate exist for $\mathcal{X}$, and if yes, what is it?
More specifically, I want to find out whether the following limit exists or not$$H(\mathcal{X})=\lim_{n\rightarrow \infty}\frac{1}{n}H(X_1,X_2,\cdots, X_n)$$ which reduces, for the present problem to $$H(\mathcal{X})=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}H(X_k)$$ My approach to the problem is if I can find that the sequence $\{H(X_n)\}_{n\in \mathbb{N}}$ converges to some limit $L$, then by Ces$\grave{a}$ro mean theorem, I have that $$H(\mathcal{X})=L$$ Now the question is, how to find whether the sequence $\{H(X_n)\}_{n\in \mathbb{N}}$ converges or not?