## sampling from transition equation (state space model)

Hi.

Consider the following state space equation
$$x_k = f(x_{k-1},v_k)$$
where f(.) is a nonlinear function and v_k is a white Gaussian noise (independent of state).

I want to generate samples according to $$p(x_k | x_{k-1} = x_{k-1}^i)$$ for a particular $$x_{k-1}^i$$. I've seen people generate samples $$v_k^j$$ from $$p(v)$$ (which is a Gaussian) and consider $$x_k^{(i,j)} = f(x_{k-1}^i,v_k^j)$$ as samples generated according to $$p(x_k | x_{k-1} = x_{k-1}^i)$$.
I can understand the intuition behind this approach, but can you suggest any mathematical proof?

Thanks,
K.
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 Tags non linear, sampling, state space