# Tagged Questions

**4**

votes

**0**answers

58 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 ...

**2**

votes

**0**answers

84 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 ...

**1**

vote

**0**answers

37 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 ...

**-1**

votes

**0**answers

42 views

### smallest (sub-) sigma algebra containing a null set [on hold]

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$
What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$.
I'm kind of ...

**5**

votes

**0**answers

76 views

### An Interesting Markov Chain Mixing time question (coupling of random variables)

Consider a simple markov chain called reproduction process. Let $N>0$ be fixed. At each time $t$, there are $N$ balls, in which $x$ balls are colored white, and $N-x$ balls are colored black. We ...

**-2**

votes

**2**answers

92 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

72 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 ...

**3**

votes

**1**answer

155 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 ...

**0**

votes

**0**answers

169 views

### Is this topological looking space metrizable? [on hold]

First of all I am sorry if something is missing or if I am making a mistake. I am a bit new in this field. The question is as follows:
The topology that is considered is called gross error model, or ...

**1**

vote

**1**answer

110 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $rank(A)=r=2^{O(\sqrt{\log_2n})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
$$\lim_{n\rightarrow\infty}\mathsf{P_{A\in\{0,1\}^{n\times ...

**2**

votes

**0**answers

62 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 ...

**-2**

votes

**0**answers

31 views

### calculating E(Xt^2,Xt-h^2) with Xt normal(0,sigma^2) [on hold]

the problèm is
let {X_t} all normal N(0, sigma^2)
défine rho_X(h)=cov(X_t,X_t-h)/var(X) and Y_t=(X_t)^2
proove that rho_Y(h)=[rho_X(h)]^2
i know that i have to use expected value E(E(Y/X)) but i don't ...

**1**

vote

**1**answer

69 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

48 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

62 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 ...

**0**

votes

**1**answer

130 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

72 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

167 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

38 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

37 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

22 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

70 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 ...

**3**

votes

**0**answers

44 views

### Bounds on truncated empirical mean

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables, $|X_i|\leq1$ and $S_n=\frac{1}{n}[X_1+\cdots+X_n]$.
Is there an upper bound on the probability of ...

**2**

votes

**0**answers

107 views

+50

### 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

96 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

**0**answers

26 views

### Asymptotic blocking probability [closed]

Intuitively, when the load of a queue $\rho<1$, the asymptotic blocking probability approaches zero as buffer size $N$ goes to infinity. Were there any conclusions on this?

**3**

votes

**1**answer

134 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

65 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

131 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 ...

**-1**

votes

**0**answers

87 views

### Martingales and Bipartite graphs

Would a vertex exposure martingale be useful for bounding the deviation in size of the largest matching from it's expected value in the standard random bipartite graph with vertex classes of size $n$ ...

**0**

votes

**0**answers

29 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 ...

**-1**

votes

**0**answers

74 views

### Stochastic ordering preserving operators on $n$ random variables

This question is about the operators which can preserve stochastic ordering.
Let $\mathcal{L}:\mathbf{X}\rightarrow \mathbb{R}$ be an operator where $\mathbf{X}=[X_1,X_2,...,X_n]$ and consider two ...

**3**

votes

**1**answer

100 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

46 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

106 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 ...

**3**

votes

**0**answers

119 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. ...

**2**

votes

**2**answers

131 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

53 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

27 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

**1**answer

90 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 ...

**4**

votes

**0**answers

51 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

279 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

64 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

82 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

110 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 ...

**3**

votes

**0**answers

110 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

**3**answers

131 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 ?

**1**

vote

**0**answers

103 views

### Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?

Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ ...

**0**

votes

**1**answer

72 views

### reverse FKG type inequality for slightly correlated Gaussian vectors

Let $X$ be a $m$-dimensional Gaussian vector, and $Y$ a $q$-dimensional Gaussian vector, for some $m,q\geq 1$. Assume that the $X_i$ and $Y_j$ are centred and have unit variance. Assume that $E X_i ...

**1**

vote

**0**answers

54 views

### Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where ...