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I have a homogenous, discrete time Markov process, $(X_n)_{n\geq 0}$, with state space $\mathbb R_+$. Its transition probabilities have a density, $f(x_n\mid x_{n-1})$, with respect to the measure $\lambda+\delta_{x_{n-1}}$, where $\lambda$ is the Lebesgue measure and $\delta_{x_{n-1}}$ is the Dirac measure that puts all the weight on the previous point, namely $x_{n-1}$.

Does there exist a measure, $\mu$, such that $(X_1, \dots, X_m)$ has density with respect to $\mu$? How do I calculate such density?

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  • $\begingroup$ The answer is in the question: the dominating measure is the product of the measures $\lambda+\delta_{x_{n-1}}$. $\endgroup$
    – Xi'an
    Apr 26, 2016 at 9:34
  • $\begingroup$ @Xi'an hmm.. I don't think the product will be a measure on $\mathbb R_+^m$ because it seems that the product will depend on the event, $\omega$. $\endgroup$
    – svendvn
    Apr 26, 2016 at 10:40
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    $\begingroup$ If the initial draw is from a continuous distribution, then none of the marginals have delta functions: they get smoothed out when you integrate over $X_1$. The joint density, though, has positive mass on the "diagonal" subspaces $X_1=X_2$, $X_2 = X_3$, etc., and on all of their intersections. If it makes you feel more comfortable, then write $g(x_1, x_2) = \delta_{x_1}(x_2)$ and write the joint density in terms of $g$. You're certainly allowed to evaluate the joint density or the dominating measure differently at every value of $X$ (within reason). $\endgroup$ Oct 4, 2016 at 1:39

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