# Tagged Questions

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### Probability densities and Absolute continuity

I've not deep knowledge in measure theory/real analysis but just few concepts given me during this second year probability course. I'm trying by myself to understand more, but I don't want to dive in ...
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### A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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### Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
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### NUMBER OF ATOMS IN A SIGMA-ALGEBRA [on hold]

I have been trying to solve the followIng question. DESCRIBE THE SMALLEST SIGMA ALGEBRA CONTAINING 'n' ARBITRARY SUBSETS OF THE SAMPLE SPACE.GIVE AN UPPER BOUND FOR THE NUMBER OF SETS IN THIS SIGMA ...
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### Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
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### Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
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### Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
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### Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
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### convergence in measure of min $(f_n,g)$

I was reading a proof of a convergence in measure variant of fatou's lemma earlier and there was a seemingly easy part of it I just could not verify. Assume $(f_n)_{n \in \mathbb N}$ is a sequence of ...
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### Almost Trivial $\sigma-$fields

I am trying to understand the proof of the following Lemma form the book A probability path by sidney Resnick. Lemma: Let $\mathcal{G}$ be an almost trivial $\sigma-\text{field}$ and let $X$ be a ...
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### Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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### How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...