1
vote
0answers
34 views

Chernoff Binomial Bound

I am reading a paper and the following Chernoff-type bound is presented: For X~Bin(n,p) and a>0, the following bounds for lower and upper tail, respectively, hold: $$\Pr[X\le np-a]\le ...
3
votes
1answer
176 views

Mathematician who talked about the probability of a “good” graph?

In my undergraduate years, one of my professors always talked about this one mathematician who was talking about "good" graphs and wondered about the existence of such a graph. Apparently this ...
1
vote
0answers
58 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
1
vote
2answers
26 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
1
vote
0answers
30 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
0
votes
1answer
30 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
1
vote
0answers
27 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
0
votes
0answers
28 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
0
votes
0answers
15 views

Searching for theorems that prove almost sure convergence from convergence in probability

As we can see that almost sure convergence implies convergence in probability, and the converse is not necessarily true. But now I would like to prove a particular sequence of random variable ...
2
votes
0answers
45 views

Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
1
vote
1answer
30 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
0
votes
1answer
36 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
1
vote
0answers
23 views

Limit a.s. of a sequence of normal random variables is normal.

I know that the statement "If $X_n$ is a sequence of normal random variables which converges a.s. to a random variable $X$, then $X$ is also a normal random variable" is true. However, do you ...
0
votes
0answers
51 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
2
votes
2answers
30 views

Markov property for a Stochastic Process

My question: Every Stochastic Process $X(t), t\geq 0$ with space states $\mathcal{S}$ and independent increments has the Markov property, i.e, for each $\in \mathcal{S}$ and $0\leq t_0\leq< ...
2
votes
1answer
142 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
1
vote
1answer
32 views

Kolmogorov's Existence Theorem

My analysis professor told us to take the following theorem for granted in order to prove other results, but I would like to see a proof of it, since I think it will be beneficial. Here is the ...
4
votes
1answer
47 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
1
vote
1answer
46 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 ...
1
vote
2answers
74 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 ...
8
votes
2answers
219 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
134 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
111 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
77 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
55 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
30 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
48 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 ...
3
votes
4answers
49 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
171 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 ...
3
votes
1answer
108 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
66 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
53 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
57 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
36 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 ...
4
votes
1answer
233 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
54 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
72 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!
5
votes
1answer
139 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
133 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
21 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
71 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
47 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
113 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
91 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
106 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
196 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
95 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
63 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
59 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 ...