Radon–Nikodym Derivatives for Dependent Sequences

Summary

The document asks how to express the Radon–Nikodym derivative between two probability measures on the sigma-field generated by a sequence when the variables are dependent. It first states the familiar independent and identically distributed case: the finite-sequence likelihood ratio is the product of the one-variable density ratios. It then asks what replaces that product when independence fails.

No answer or derivation is included, so the document does not provide a formula for dependent sequences or establish conditions under which the measures are mutually absolutely continuous. The question points toward the general principle that a joint likelihood ratio, or a product of conditional likelihood ratios when such conditional densities exist, depends on the sequence’s dependence structure. That principle is context for understanding the question, not a result demonstrated in the text. The material is a probability-theory prompt with possible relevance to change of measure in quantitative finance, rather than a trading strategy or empirical analysis.

Key ideas

• For independent identically distributed observations, the finite-sample likelihood ratio factors into one-variable density ratios.
• Dependence means that marginal density ratios alone do not generally determine the joint change of measure.
• The document asks how to represent the derivative on the sigma-field generated by dependent observations.
• It provides no answer, assumptions, or worked example for the dependent case.

Tags

# The Radon-Nikodym derivative for a sequence of dependent variables


# The Radon-Nikodym derivative for a sequence of dependent variables












Suppose that a probability space $(\Omega, \Sigma, \mathbb{P})$ is given. Let $W=\{W_n\}_{n\in \mathbb{N}_0}$ be a sequence of $\mathbb{P}$-i.i.d real-valued random variables on $\Omega$. Furthermore, assume that $\mathbb{Q}$ be a new probablity measure on $\Sigma$, and $W$ is $\mathbb{Q}$-i.i.d and that $\mathbb{Q}_{W_1} \sim \mathbb{P}_{W_1}$. We denote by $\mathcal{F}^{W} = \{\mathcal{F}_n^W\}_{n\in \mathbb{N}}$ the natural filtration of $W$. Then, one can say that for every $n\in \mathbb{N}_0$ and for all $D\in \mathcal{F}_n^W = \sigma(W_1, W_2, .., W_n)$ \begin{equation} \mathbb{Q}(D) = \mathbb{E}^{\mathbb{P}}\Big[I_D \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j)\Big] \end{equation} or equivalently, \begin{equation} \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W} = \prod_{j=1}^{n}\frac{d\mathbb{Q}_{W_1}}{d\mathbb{P}_{W_1}}(W_j) \end{equation} Now, my question is what if $W_j's$ is not independent. In this case, how does the Radon-Nikodym derivative $\frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_n^W}$ look like?

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