Thinning a Renewal Process – Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a general Renewal point process? i.e.,

If in a renewal point process with rate $\lambda$, we keep each point with probability $p$ independently, do we obtain another renewal point process?

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Suppose $\{X_{n,k}\}$ are i.i.d. with mean $\lambda^{-1}<\infty$ and $\{\gamma_n\}$ are i.i.d. with $\mathsf{Geo}(p)$ distribution. Then the sequence $$Y_n := \sum_{j=1}^{\gamma_n} X_{n,j}$$ is i.i.d. and thus defines a renewal sequence with mean $$\mathbb E[Y_1] = \mathbb E[X_{1,1}]\mathbb E[\gamma_1]=(\lambda p)^{-1}$$
(by Wald’s identity), or equivalently “rate” $\lambda p$.