# Tagged Questions

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### Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
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### How we compute expectation of a singular random variable?

In probability (or measure) courses, we often see the Cantor distribution that is singular with respect to the Lebesgue measure. Its CDF is increasing but whenever its differentiable, the ...
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### A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
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### Where does this probability problem come from?

A long time ago, a friend gave me a probability problem. Here is rough reconstruction. A spaceship is lost in deep ($3$-d) space. Its home planet is $X$ meters away. Every second, the spaceship ...
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### A list of different measures of distance/difference/dissimilarities/similarity of two probability distributions?

I wanted to know more about the different methods for comparing the similarities of two probability distributions P and Q. I wanted a list of the different methods that exist for comparing ...
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### Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
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### Name/significance of integral of the square of a probability density function

Background/Motivation Given a probability density function $f(x)$, the mean of the corresponding random variable is the $x$-coordinate of the centroid of the region under the graph of $f$. I ...
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### Convergence Theorems for Random Variables

I am trying to better understand the notion of convergence for a sequence of random variables. I was looking into "Probability and Measure"- Billingsley but in this book the link to the convergence ...
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### Exposition of Erdős and Rényi's 'New law of large numbers'.

Where can I find an exposition of the paper On a new law of large numbers by Erdős and Rényi? I'm reading this paper and it's rather terse, so I'd like some intuition and explanation. I did a Google ...
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### Reference Request: Video Lectures for Stochastic Processes

It is difficult to learn Stochastic Process by self-reading. Can you provide some video lectures on Stochastic Process?
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### A Good Book for Mathematical Probability Theory [duplicate]

I am from mathematical background, and I always hated the way they teach elementary probability theory in schools without giving any clue about measure theory. I want a theoretical book in ...
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### Boundedness of expected reward Markov chain (may be related to discret $M/M/\infty$ queue)

[EDIT]: I read a bit on $M/M/\infty$ queue and it may not be the right comparison and my notation may be confusing (I'm in discrete time and $\lambda,\mu$ look likes rates when they are probability). ...
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### Reference request - maximum conservative extension of a probability assignment

I made up some definitions which probably lead to some interesting mathematics. However, I suspect they've studied before. So that I don't end up reinventing the wheel (always bad!), I'm after a ...
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### Books at similar levels as Kallenberg' Foundation of Modern Probability?

Thanks to many people who have mentioned it to me and others on this site before. I was just able to peek into Kallenberg' Foundation of Modern Probability. It is more comprehensive, deep and thorough ...
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### “Interpolation” of polynomials

I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli ...
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### Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
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### Bayesian Inference in Measure Theory

What's the deal. How does this work, or can you point me to some references? I tried $\mu(A|B) = \mu(A \cap B) / \mu(B)$ and got stuck on $\mu(B) = 0$. Edit: Sorry for being lazy. My background is ...
Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...