1
vote
0answers
24 views

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 ...
0
votes
0answers
19 views

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 ...
7
votes
2answers
173 views

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 ...
2
votes
1answer
128 views

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 ...
5
votes
3answers
75 views

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 ...
1
vote
1answer
51 views

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 ...
3
votes
1answer
37 views

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 ...
0
votes
0answers
20 views

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 ...
2
votes
0answers
28 views

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 ...
1
vote
1answer
32 views

Reference for a proof of which 2-increasing functions are joint cdf's

Can somebody give me a reference giving the detailed statement and proof of the fact that the joint cdf's of positive Borel measures $\mu$ on $\mathbb{R}^2$, so $$F(a,b) = \mu(\{(x,y) : x \leq a, y ...
2
votes
4answers
33 views

Reference request for stochastic process and applications

I am looking for a text book that will cover the following topics I hope someone could suggest me a good text book that will provide me a good guidance regarding the following; Generating functions, ...
2
votes
1answer
110 views

What are some good books about martingales?

I'm looking for suggestions for well written books dealing with martingale theory, not necessarily exclusively. I'm also looking for a nice compilation of problems, preferably with answers, on this ...
2
votes
1answer
69 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
3
votes
0answers
47 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...
1
vote
1answer
49 views

Strong Markov property given transition functions

Suppose we are given family of transition functions satisfying Chapman-Kolmogorov equation, what conditions will ensure that there exists a continuous or cadlag Markov process with given transition ...
1
vote
1answer
50 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
1
vote
1answer
30 views

A clear reference on regular conditional distributions?

I've been trying to learn about regular conditional distributions from Klenke's book on probability theory, but I'm incredibly confused. I looked at Durrett's book, but his chapter on regular ...
3
votes
1answer
127 views

Reference on Doob's h-transform

I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly ...
3
votes
0answers
51 views

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers ...
1
vote
0answers
61 views

Orthogonality of the Hermite polynomials: probabilistic approach

Can anyone help me with the following question: Is there any reference in which a probabilistic approach was used to prove that the Hermite polynomials are orthogonal? Thanks a lot!
4
votes
1answer
108 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
1
vote
3answers
114 views

Which is a good textbook on stochastic processes which takes measure theoretic approach?

I was looking for an intermediate-advanced textbook on stochastic process. I have graduate level probability knowledge.
0
votes
0answers
20 views

Establishing recurrence and positive recurrence of Markov processes via “barriers”?

I've been reading the book by Wentzell and Freidlin on dynamical systems with small random perturbations. On page 42 it's stated: It is possible to give stronger conditions for recurrence and ...
1
vote
1answer
59 views

Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
2
votes
2answers
40 views

A Reference Book Justifying Different Distributions

Well I am trying to find a book that could come up with a rationale behind different distributions but not only defining them and giving an intuition about the structure of distributions. For example ...
4
votes
1answer
98 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
2
votes
0answers
85 views

Textbook Recommendation; Proability Theory with Measure Theory

I'm currently taking a course in Probability Theory and was hoping someone could point me in the direction of a useful supplementary textbook. Our course currently uses A Modern Approach to ...
1
vote
0answers
87 views

A query on Palm Khintchine Theorem's proof

I was searching for a good reference on Palm Khintchine theorem proof. When I googled it, I got the following reference (as a Google book) here. It states that a superposition of independent "low ...
2
votes
0answers
164 views

Applying a linear operator to a Gaussian Process results in a Gaussian Process: Proof

In this paper, it is stated without proof or citation that "Differentiation is a linear operation, so the derivative of a Gaussian process remains a Gaussian process". Intuitively, this seems ...
3
votes
1answer
91 views

Measure dualization

What ways are known to correspond, or transfer, a Borel probability measure $\mu$ over some Banach space $X$ to a Borel probability measure $P$ over $X^{*}$, the dual space? Of course, if $X^{*} = ...
2
votes
0answers
55 views

almost sure convergence of sums of triangular arrays

A well known result (see for example Kallenberg Theorem 4.17) is that if $x_j$ are symmetric independent random variables, then the following are equivalent: i)$\sum x_j<\infty$ almost surely; ...
3
votes
1answer
57 views

Lebesgue density for other probability measures on $[0,1]$

Does the Lebesgue density theorem hold for arbitrary (Borel) probability measures on $[0,1]$? Following Downey & Hirschfeldt's proof leads me to believe that the answer is "yes". (Recall every ...
0
votes
0answers
40 views

Estimate a level set of the form $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$

Suppose I have a continuous function $f(\mathbf{x}):\mathbb{R}^d \mapsto\mathbb{R}$. I am interested in the level set $A \equiv \{\mathbf{x} \mid f(\mathbf{x})=\alpha \}$. Suppose the lebesgue measure ...
5
votes
1answer
130 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
1
vote
0answers
52 views

Weak Law of Large Numbers proof

I want to know if there is a proof of the Weak Law of Large Numbers without using the Chebyshev's Inequality? please can anyone give me some references
1
vote
0answers
125 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
4
votes
2answers
92 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1
vote
0answers
58 views

What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?

It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
0
votes
1answer
32 views

Convergence of characteristic functions for random vectors: references?

Let $(X_n)_{n\in\mathbb{N}},\ (Y_n)_{n\in\mathbb{N}}$ two independent sequences of real random variables. Suppose $X_n\xrightarrow[n\to\infty]{d}X,$ $Y_n\xrightarrow[n\to\infty]{d}Y$. Then I'm quite ...
6
votes
1answer
169 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
0
votes
1answer
467 views

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?
5
votes
3answers
442 views

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 ...
2
votes
0answers
152 views

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). ...
1
vote
0answers
76 views

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 ...
3
votes
1answer
237 views

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 ...
2
votes
3answers
105 views

“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 ...
4
votes
1answer
360 views

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 $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
4
votes
1answer
252 views

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 ...
3
votes
2answers
192 views

Generalization of a product measure

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 ...
2
votes
3answers
135 views

Is all of probability theory “derivable” from Bernoulli trials?

I once heard an offhand remark that all of probability theory is "derivable" (whatever that means) from Bernoulli trials. Is there any truth to this statement? I might imagine that, say, all ...