For questions involving the notion of the Radon-Nikodym derivative or the Radon-Nikodym theorem. Use this tag along with (probability-theory) or (measure-theory).
Questions tagged [radon-nikodym]
384 questions
54
votes
3 answers
Intuition for probability density function as a Radon-Nikodym derivative
If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all $x \in \mathbb{R}$.
More rigorously, I could…
bcf
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Radon–Nikodym derivative and "normal" derivative
The Radon–Nikodym theorem states that,
given a measurable space $(X,\Sigma)$, if a $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ is absolutely continuous with respect to a $\sigma$-finite measure $\mu$ on $(X,\Sigma)$, then there is a measurable…
Tim
- 49,162
15
votes
1 answer
What is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure?
The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $\mu$ is essentially defined by a collection of…
fromscratch
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13
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12
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Computing Radon-Nikodym derivative
I learned Radon-Nikodym theorem in class and I know what exactly it is.
But I am not sure about how to compute Radon-Nikodym derivative... Any reference does not explicitly say about how to compute Radon-Nikodym derivative..
Can anybody help me…
Detectives
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11
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2 answers
Rigorous definitions of probabilistic statements in Machine Learning
In a supervised machine learning setup, one usually considers an underlying measurable space $(\Omega, \mathcal{F}, \Bbb P)$ and random vectors/variables $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \rightarrow \Bbb R.$ We can then consider the…
John D
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votes
1 answer
Distributional derivative of measures and Radon-Nikodym derivative.
The distributional derivative of a function $f$ is the function $g$ that verifies $\int f\phi'=-\int g\phi$. In particular, if f is differentiable, the distributional derivatives is the same as the standard derivative. Now similarly, if $\mu$ is a…
edamondo
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9
votes
2 answers
Calculating Radon Nikodym derivative
I am trying to calculate the Radon-Nikodym derivative for $\mu = m + \delta_0$ where $m$ is Lebesgue measure over a compact subset of $\mathbb R$ and $\delta_0$ is Dirac measure at $0$.
Clearly, $m \ll \mu$ and $\mu \perp \delta_0$. Therefore, the…
Ram
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votes
1 answer
$G$ acts transitively on a space $X$. If a function on $X$ is $G$-invariant up to measure zero, is it necessarily a constant (up to measure zero)?
Consider a locally compact Hausdorff $σ$-compact topological space $X$ and a locally compact Hausdorff $σ$-compact topological group $G$ acting continuously and transitively on $X$ such that there exists a $G$-invariant Radon measure $\mu$ on…
Carlos Esparza
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9
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1 answer
Finding Radon-Nikodym derivative
Let $m$ be Lebesgue measure on $\mathbb R_+=(0,\infty)$ and $\mathcal A = \sigma\left(( \frac 1{n+1} , \frac 1n ]:n=1,2,...\right)$. Define a new measure $\lambda$ on $\mathcal A$, for each $E \in \mathcal A$, by $\lambda(E)= \int_E fdm $, where…
bellcircle
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8
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1 answer
Does Radon-Nikodym derivative affect the Variance of a Random Variable?
Edit 26 January 2022: The answer below elegantly shows that when the diffusion term of an Ito process is not constant in $\omega$, it is generally not true that Variance remains unaffected by the change of measure.
In the question below, however,…
Jan Stuller
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8
votes
2 answers
Conceptual Issues in the Measure Theoretic Proof of Conditional Expectations (via Radon-Nikodym)
I have been looking into measure theory (from a probabilist's perspective), and I have found the proof of the existence of the conditional expectation to feel a little "glossed over" in literature. As such I've tried to very, very slowly break down…
tisPrimeTime
- 942
8
votes
1 answer
Radon-Nikodym-derivative as a martingale
At the beginning of all the stuff about Girsanov theorem, we introduced the Radon-Nikodym derivative as $Z_\infty := \frac{d \mathbb{Q}}{d \mathbb{P}}\vert_{\mathcal{F}_\infty}$.
Next, we considered the following:
\begin{align}
Z_t =…
tubmaster
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8
votes
1 answer
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in [0,T]}$ is standard $\mathbb P$-Brownian…
BCLC
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Has Mathematical Finance education become unnecessarily inaccessible?
Background
I had a very good and bright student with some decent exposure to markets and the standard college math curriculum who got overwhelmed during a mathematical finance course due to unnecessary off-topic formalisms. In particular, the…
Pellenthor
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