**3**

votes

**1**answer

148 views

### Lower bound for the $p$-th absolute moment of a sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $\operatorname E X_k=0$ and $\operatorname E |X_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the ...

**0**

votes

**3**answers

190 views

### An inequality based on expectation of continuous random variables

I am trying to prove the following statement:
$$
E[g(X)] E[X^2g(X)]\ge E[Xg(X)] E[Xg(X)]
$$
where $X$ is a random variable, $E[\cdot]$ denotes the expectation operator with respect to ...

**0**

votes

**0**answers

32 views

### Reference request for specific POMDP examples

Following is strictly for discrete-time discrete-space Markov chain.
Consider a partially observed Markov decision process (POMDP) $P = \{X,O,A,P,B_a\}$.
Here $X = \{x_1, \cdots, x_n\}$ refers to ...

**5**

votes

**3**answers

121 views

### Random partitions with prescribed pairwise membership probabilities

Let $(p_{ij}) \in [0,1]^{n \times n}$ be a given symmetric matrix, with $1$ on the diagonal. Suppose $\pi$ is a partition of $[n]=\{1,\dots,n\}$ and let us write $i \stackrel{\pi}{\sim} j$ if $i$ and ...

**3**

votes

**1**answer

255 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

**5**

votes

**0**answers

102 views

### What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...

**0**

votes

**1**answer

189 views

### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...

**2**

votes

**0**answers

51 views

### minimal entropy approximation of a discrete random variable

Let $X$ be a $\mathbb{N}$-valued random variable. Define
$$
H^\epsilon_n(X) = \inf_f H(f(X))
$$
where $f$ runs over all functions $\mathbb{N} \to \mathbb{N}$ such that $\Pr(f(X)\neq X)<\epsilon$ ...

**1**

vote

**0**answers

55 views

### Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation
of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$.
Let $S(n)$ be ...

**7**

votes

**0**answers

140 views

### Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers).
More generally, suppose we fix any ...

**5**

votes

**1**answer

264 views

### Can a Brownian motion be fast at its extrema?

After pondering this MO question > Location of maximum of Brownian motion with rough drift <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its ...

**2**

votes

**0**answers

106 views

### A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...

**0**

votes

**1**answer

236 views

### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where ...

**-2**

votes

**2**answers

120 views

### Equality of two conditional expectations

I would like to show that for any random variable $X$ and $Z$ such that $X$ and $Z$ are independent and for any measurable functions $f$ and $g$,
$$ \mathbb E \left[ f(g(X),Z) | g(X) \right] = ...

**1**

vote

**0**answers

139 views

### Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...

**5**

votes

**1**answer

316 views

### Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...

**1**

vote

**1**answer

187 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
...

**2**

votes

**0**answers

83 views

### Convergence in distribution of stochastic equation solutions

I post this post en MSE (link) but I think that is more suitable for this site.
I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the ...

**1**

vote

**1**answer

112 views

### 1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...

**1**

vote

**1**answer

122 views

### Discrete Maximum Entropy Distribution with given mean

For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As ...

**1**

vote

**1**answer

94 views

### Conversion between condtional expection conditioned on $\sigma$-algebra and on r.v

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\mathcal G \subseteq \mathcal F$ be a sub-$\sigma$-algebra of $\mathcal F$ and $X : \Omega \to \mathbb R$ a random variable. Then the ...

**1**

vote

**1**answer

206 views

### Book on Convergence Concepts in Probability without Measure Theory [closed]

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...

**4**

votes

**1**answer

109 views

### On Minkowski sum of two independent Poisson point processes

Suppose that $\Phi_1$ and $\Phi_2$ represents two independent Poisson point processes respectively with intensity $\lambda_1$ and $\lambda_2$ (therefore). We know very well different operations on ...

**0**

votes

**4**answers

236 views

### Approximating an arbitrary $\sigma$-algebra by simpler $\sigma$-algebras

A $\sigma$-algebra $\mathcal F$ over $\Omega$ is generated by an countable partition if there exits a countable partition $\mathcal B = \{ B_i \}$ of $\Omega$ such that $\mathcal F = \sigma(\mathcal ...

**0**

votes

**0**answers

61 views

### Generating alternating cycles on a perfect matching

Given a perfect matching $M$ in a regular bipartite graph $G$, is there an efficient algorithm to randomly generate self-avoiding alternating cycles with uniform distribution? Ideally, such an ...

**0**

votes

**0**answers

42 views

### Integral over a point process. Asymptotic of the dispersion

I consider an integral (or a sum with random index)
$$
M(t) =\int\limits_0^t f(t-u)dX(u),
$$
where
$$
X(u) = \sum\limits_{i=1}^{N(u)} \xi_i,\qquad N(u)=\max\{k: \tau_1+\,\dots,\,\tau_k\, <\, u\},
...

**0**

votes

**0**answers

28 views

### Moments in the Quantile Process

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F:
$$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$
In this case, $F^{-1}$ is the (generalized) inverse of ...

**0**

votes

**0**answers

78 views

### Discrete measures and discrete kernels

This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...

**2**

votes

**0**answers

117 views

### Probabilities involving Beurling density

I am interested in calculating probabilities involving Beurling densities. Since it's likely probabilists are not familiar with the definitions, I give them below.
Definitions.
A metric space is ...

**2**

votes

**0**answers

172 views

### Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...

**3**

votes

**1**answer

169 views

### Equivalence of Gaussian measures

Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv ...

**1**

vote

**0**answers

104 views

### Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...

**1**

vote

**1**answer

228 views

### Problem on convergence in probability measres

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have
$$\int fd\mu_n \to \int fd\mu$$
Then I have seen in some place ...

**0**

votes

**0**answers

37 views

### Bahadur-Kiefer representation and KMT embedding

I am interested in the connection between the so called Bahadur-Kiefer process and the KMT/Hungarian embedding. At first sight there seems to be a relationship between the topics, but oddly enough ...

**3**

votes

**1**answer

229 views

### Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = ...

**3**

votes

**1**answer

123 views

### Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable.
Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria:
...

**1**

vote

**0**answers

62 views

### Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...

**2**

votes

**1**answer

123 views

### Ergodicity for the mean of a linear process without finite second moment

Suppose that $\{X_k:k\in\mathbb Z\}$ is a linear process, i.e. a sequence of random variables such that
$$
X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}
$$
for each $k\in\mathbb Z$, where ...

**6**

votes

**1**answer

261 views

### Location of maximum of Brownian motion with rough drift

I am interested in the distribution of the $\text{argmax}_{t \in [0,1]} \{B(t) + f(t)\}$, where $B$ is a Brownian motion (or Brownian bridge) and $f:[0,1] \to \mathbb{R}$ is a continuous function. ...

**1**

vote

**2**answers

257 views

### Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...

**1**

vote

**0**answers

57 views

### Probability that top k elements will be got?

Suppose $n$ vaiables $a_1$ ~ $a_n$ distributed over $[1,n]$. Without loss of generity, let $a_i =n- i+1$.
Suppose that each variable has a probability $p_e$ to be wrong, and the wrong value uniformly ...

**0**

votes

**0**answers

44 views

### Restricted singular values of Wishart matrices

This is an extended question of
Restricted singular values of random matrix.
It is well-known that the smallest singular value of a $p \times \frac{p}{2}$ matrix consisting of i.i.d. ...

**2**

votes

**2**answers

228 views

### Uniqueness in martingale representation theorem

Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is ...

**5**

votes

**0**answers

70 views

### Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...

**13**

votes

**1**answer

324 views

### How to roll a $p$

Let $p$ be a positive integer (which is not a power of $2$), and suppose we want to generate a number uniformly randomly in the set $\{ 0, 1, \dots , p-1 \}$ (to emulate a dice roll). We are given ...

**1**

vote

**1**answer

80 views

### Concentration bound for a martingale-like setting (the expected difference decreases as the sequence increases)

I went through several martingales concentration bounds, but none of them fit the settings I am interested in, which is the following. Suppose I have a sequence of nonnegative random variables ...

**1**

vote

**0**answers

115 views

### Is the stationary distribution of this Markov chain uniform?

First, a little bit of background: Since 2012, Canada has decided to phase out the penny for its coinage system. Product prices may still use arbitrary cents, especially since prices do not typically ...

**2**

votes

**1**answer

149 views

### Problem on convergence in space of probability measures

Let $\mu_n, \mu$ be a sequence of probability measures on a Polish space $S$ and $\mu_n', \mu'$ be some kind of extension of $\mu_n, \mu$ on $\bar{S}$ such that all the boundary points of $S$ gets a ...

**4**

votes

**1**answer

224 views

### Can the method of small moments prove a bound on the norms of random trilinear forms?

If $F(v_1,\dots,v_k)$ is a $k$-linear form on $\mathbb R^n$, the norm I want to consider is
$$ ||F|| = \sup \frac{ F(v_1,\dots, v_k)}{\prod_{i=1}^k \left|\left|v_i\right|\right|} $$
where the vector ...

**1**

vote

**2**answers

157 views

### spaces of probability measures on a Polish space and the convergence

I want to read the topic "spaces of probability measures on a Polish space and the convergence". What is the best reference for that ?