# Tagged Questions

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59 views

### Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...

**1**

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**0**answers

136 views

### Order statistics of iid uniform RV and Pólya's urn model. Question about a.s. convergence

Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let
$$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$
be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there ...

**1**

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**0**answers

43 views

### Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution ...

**1**

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**2**answers

126 views

### Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables.
Are there cases where one can been able to quantify $...

**6**

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**1**answer

109 views

### Basic Definition and Notations in RWRE

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...

**0**

votes

**1**answer

90 views

### Exact formula for computing n-step transition probability of random walks with self-transitions

Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$...

**4**

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**2**answers

222 views

### Variation of Radon transform for probability measures on $\mathbb C$

Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...

**-1**

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**0**answers

97 views

### Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...

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**0**answers

99 views

### A two variable recurrence relation with conditionals

I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence
$$
f(n,m) = \begin{cases} f(n, \...

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**4**answers

504 views

### Speed of convergence in Lebesgue's density theorem

Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim_{\epsilon\...

**7**

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**0**answers

143 views

### Winding number of a random walk on the square lattice before hitting the origin

Let us consider a simple random walk on $\mathbb{Z}^2$ started at $(x,0)$ and killed upon hitting the origin. Define the total winding number $w_x$ around the origin to be the (signed) number of ...

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**0**answers

67 views

### Law of large numbers for random functions?

Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n =...

**1**

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**0**answers

25 views

### Reference Request: $M_t/M_t/1/K$ queue length distributions

I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that ...

**3**

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**1**answer

328 views

### What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...

**0**

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**0**answers

76 views

### Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf),
Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$
be a family of functions ...

**2**

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**1**answer

95 views

### Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$
...

**0**

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**0**answers

73 views

### Laplace transform (or characteristic functional) of atomic random measure

A random (nonnegative Radon) measure $M$ (on $\mathbb R^n$, say) has its law characterized by the Laplace transform $\mathbb E\exp(-\int \varphi(x)\ M(dx))$, $\varphi\in C_c^+(\mathbb R^n)$ (...

**5**

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**0**answers

92 views

### Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...

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**3**answers

713 views

### On the sum of uniform independent random variables

Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$?
...

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**0**answers

46 views

### Two dimensional Brownian motion moving from one point to other

Suppose $W_t= (X_t,Y_t)$ is a $2$d standard Brownian motion starting at $(-1,0)$. How do I show that there is a positive probability that $W_t$ moves from $(-1,0)$, to a neighborhood of $(1,0)$, say ...

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80 views

### Brownian hitting probability of a small body

Consider a Brownian motion $B(t)$ starting from the origin $0$ in $\mathbb{R}^n$. Consider the ball $B(0, r)$ and an open set $V \subset B(0, r)$. If it is known that the probability of the Brownian ...

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**0**answers

67 views

### Brownian motion in perturbed (asymptotically flat) metric

Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...

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**0**answers

49 views

### Criterion for weak compactness (from Revuz and Yor)

I'm trying to read a proof in the book or Revuz and Yor (prop 1.5 Chapt 13, p.516-517, "continuous martingales and Brownian Motion" 3rd edition), but I'm struggling with a detail. I think it lies ...

**4**

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**1**answer

272 views

### A variant of bin-and-ball problem

We have $n$ balls, each belonging to a group (e.g, color). There are $g$ groups ($g$ may be large but $g=o(n)$). We sequentially put the balls into $m$ bins in the following way: for each ball, we ...

**3**

votes

**1**answer

101 views

### On the spectrum of stationary Gaussian process

What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian ...

**2**

votes

**1**answer

63 views

### Expectation of a function of sorted uniform random variables

Let $a,b,c>0$ with $a>b$ and let $X_1,\dots,X_n$ be a collection of independent uniform samples from the unit interval. Assume that we re-order the samples so that $X_1\leq X_2 \leq \cdots \leq ...

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**1**answer

63 views

### Uniqueness of invariant measure for equivalent transition probabilities

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...

**4**

votes

**1**answer

197 views

### Collecting stones in n buckets

There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...

**6**

votes

**1**answer

220 views

### Density of convolution

Let $\{X_i\}$ be i.i.d random variables uniform on a measurable, symmetric set $A$ contained in $[-1,1]$. Let $g_{n}$ be density of $X_1+\ldots + X_n$.
Question (general): Is there any non-trivial ...

**0**

votes

**1**answer

133 views

### Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...

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**0**answers

116 views

### Asymptotics of subgraph densities in graphons

In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...

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**0**answers

57 views

### Bounding a distribution using moments

Suppose $X$ is a non-negative random variable with bounded image. I was wondering if anybody knew of any results that could answer a question of the following type: Suppose the $n$-th moment satisfies ...

**0**

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**0**answers

35 views

### information about composite random process

I have a following composite random process
$$X_j = v_0 + 1/j^2 + Y_j + Z_j$$
where $v_0$ is a constant, $Y_j \rightarrow 0$ almost surely as $j\rightarrow \infty$ and $Z_j \sim N\big(0, \frac{a^{2j}...

**1**

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**0**answers

57 views

### Why is the classical secretary problem about ranks?

This relates here: http://math.stackexchange.com/questions/1820997/why-is-the-classical-secretary-problem-about-ranks
You want to stop optimal in a sequence of items presented sequentially, that is ...

**1**

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**0**answers

33 views

### Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...

**1**

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**0**answers

60 views

### Reason of the scaling factor $n^{2}$ in Hydrodynamic limits

In some books about hydrodynamic limits, example De Masi and Pressuti, when taking about the transition from micro to macro to get the hydrodynamic limit of some process it is mentioned that in order ...

**1**

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**2**answers

120 views

### When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is ...

**3**

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**2**answers

178 views

### Lévy measure and Lévy process

A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$
A Lévy process can be characterized by triples $(b, A, \...

**0**

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**0**answers

99 views

### Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals.
What is know from ...

**5**

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**1**answer

134 views

### Contraction of probability measures

Notations. Let $(X, \mathcal{B})$ be a separable Banach space, with its Borel sigma-algebra, $\|\cdot\|$ stands for the norm in $X$, $\mathcal{P}(X)$ - the set of all probability measures on $X$. Let ...

**5**

votes

**1**answer

194 views

### Concentration bounds on weighted sum of i.i.d. Bernoulli random variables

Let $X_1,\dots, X_n\sim\operatorname{Bern}(\frac{1}{2})$ be independent, identically distributed random variables, and $\alpha=(\alpha_1,\dots,\alpha_n)\in[0,1]^n$ a vector of non-negative weights ...

**0**

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**0**answers

110 views

### Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...

**3**

votes

**1**answer

74 views

### spatial mean of log-normal random field

Let $X(x):= \exp(Y(x))$ for $x\in D\subseteq \mathbb{R}^d$ for a domain $D$, where $Y$ is a Gaussian random field with some smooth covariance function $c(\cdot,\cdot)\colon D\times D\to \mathbb{R}$. ...

**4**

votes

**1**answer

58 views

### Uniform convergence of action of Feller semigroup with $1$ variable

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...

**0**

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**0**answers

28 views

### Approximating Minkowski Sum of 3 dimensional Convex Polytopes by Sampling

Let $P_1,P_2...P_r$ be a set of convex polytopes with $n_r$ vertices in 3 dimensions. These polytopes basically represent uncertainties of '$r$' number of 3d-points respectively in space. The global ...

**1**

vote

**1**answer

67 views

### The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...

**5**

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**0**answers

75 views

### Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...

**5**

votes

**1**answer

84 views

### Probability Brownian motion lies between $2$ functions

Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \ldots < t_J$ are time points. Let $W_t$ be a standard Brownian motion. Is it possible to further simplify the expression
\...

**2**

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**0**answers

127 views

### concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...

**4**

votes

**1**answer

121 views

### What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here.
The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...