**2**

votes

**0**answers

49 views

### Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...

**9**

votes

**1**answer

131 views

### What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdos-Renyi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Alber model
...

**5**

votes

**1**answer

174 views

### Optimisation of betting strategy

Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we ...

**6**

votes

**3**answers

155 views

### Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...

**4**

votes

**1**answer

113 views

### Time for brownian motion to cross a coordinate plane

Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the ...

**5**

votes

**0**answers

180 views

### Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...

**2**

votes

**0**answers

50 views

### Coupling Marginals of Distributions on the Sphere

Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ ...

**5**

votes

**0**answers

145 views

### Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...

**2**

votes

**0**answers

103 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**2**

votes

**1**answer

88 views

### Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.

**7**

votes

**2**answers

386 views

### $\langle X\rangle_t = t$

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...

**3**

votes

**1**answer

257 views

### inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...

**0**

votes

**0**answers

147 views

### Probability two random intervals overlap

I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows:
Given N randomly ...

**0**

votes

**1**answer

107 views

### Tail bound for a martingale

The setup is as follows.
We are given a martingale $X_0,X_1,...,X_k$. The difference $X_i-X_{i-1}$ is always between $[-1,1]$. Variance $D^2(X_i-X_{i-1}| X_{i-1})$ is something, but we can show that ...

**3**

votes

**0**answers

114 views

### Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$.
Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...

**0**

votes

**0**answers

101 views

### Uniqueness of the “Gubinelli” Derivative in the Theory of Paracontrolled Distributions

From the theory of Rough Paths it is well known that if we have a truly rough path $X$ and two controlled rough paths $(Y,Y'),(Y,\tilde{Y}')\in\mathcal{D}_X^{2\alpha}$, then we have already $Y' = ...

**3**

votes

**1**answer

119 views

### Practical bounds for the Wasserstein distance in 2 dimensions

Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let ...

**10**

votes

**2**answers

708 views

### How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...

**0**

votes

**0**answers

41 views

### Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient
$$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$
where $F$ denotes ...

**10**

votes

**5**answers

536 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...

**0**

votes

**0**answers

51 views

### Convergence of approximate quadratic variation in $L^p$

For a diffusion $X_t$, I can set
$$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$
Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...

**0**

votes

**0**answers

30 views

### Derivative of a cdf with respect to a parameter

Given two independent Random Variables $X$ and $Y$ with known distributions, I would like to know if I can say that the expression
$$
\operatorname{Pr}( f (t'+Y-X)+Y-X < z)
$$
is increasing in ...

**0**

votes

**0**answers

55 views

### Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...

**4**

votes

**0**answers

122 views

### A generalization of coupon collector problem - $\geq1$ pick per experiment

Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back.
$N=1$ is standard coupon ...

**2**

votes

**0**answers

251 views

### A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$.
Denote sets ...

**1**

vote

**0**answers

78 views

### Finding a hidden “heavy” subset of random variables

Let $X_1,\dots, X_n$ be independent non-negative random variables (with finite expectation and variance), and $0 < m < n$ be a fixed integer such that there exists a subset $S\subseteq [n]$ of ...

**3**

votes

**1**answer

182 views

### variance of the number of fixed points for a permutation group

It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...

**5**

votes

**2**answers

126 views

### Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let ...

**1**

vote

**1**answer

201 views

### Can we estimate the probability $\mathbf{P}(a-k|a - b) $ on a random graph?

Let $G=(V,E)$ be an undirected random graph such that
$V$ is the set of nodes, and $E$ is the set of edges
Assume the ground graph $G$ is sparse enough, for example, $\frac{|E|}{|V|}= c \in [10, ...

**1**

vote

**0**answers

99 views

### $\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q. Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then $\mathbb{P}(d(X,Y)>\alpha)<\beta$.
Only hints please.
...

**1**

vote

**2**answers

151 views

### Measures on Young tableaux

I have seen that on the set of Young tableau the Plancherel measure was quite natural to define. I was wondering if other measures were also studied. In particular, a simple exemple which comes to my ...

**0**

votes

**0**answers

29 views

### Consistency of M-estimators when the constraint set also has to be estimated

Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
...

**2**

votes

**1**answer

87 views

### Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...

**0**

votes

**0**answers

37 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**3**

votes

**1**answer

122 views

### 'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...

**3**

votes

**1**answer

153 views

### Matching moments in even dimensions

Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu_i=\mathbb{E}_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. ...

**0**

votes

**0**answers

58 views

### $\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that
$$
X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}
$$
for some parametric distribution $\mathcal{B}$, where ...

**6**

votes

**1**answer

153 views

### Tighter Caratheodory on the moment curve?

The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...

**3**

votes

**1**answer

61 views

### Cramer-Rao bound for randomized estimator

As is well known, the Cramer-Rao bound (or information inequality) sets a lower bound on the variance of estimators of a parameter.
Consider the case when the parameter is a scalar, the estimator is ...

**1**

vote

**0**answers

59 views

### What is a two-sided geometric distribution?

I found in some articles (such as this) references to two-sided geometric distribution. But I went through texts of probability and did not find anything called "two-sided geometric distribution". ...

**2**

votes

**0**answers

41 views

### A canonical example of the non-existence of predictive probability distribution

Section 3 of Fortini et al. (2000) states that
Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...

**0**

votes

**1**answer

38 views

### Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...

**1**

vote

**2**answers

90 views

### Is zero a regular point for a drifted $\alpha$-stable process?

We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$,
where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha
\in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$,
and ...

**3**

votes

**0**answers

120 views

### Do binary symmetric channels maximize mutual information?

Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. ...

**0**

votes

**0**answers

89 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**1**

vote

**2**answers

109 views

### Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...

**6**

votes

**1**answer

158 views

### Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...

**0**

votes

**0**answers

33 views

### Distance minimization and submodular functions

I have a question about square distance minimization and submodular functions. Suppose I have a random variable $X = (X_1,...,X_n)$ over $\mathbb{R}^n$. Let $x$ be a realization of this random ...

**5**

votes

**2**answers

218 views

### Infimum of Gaussian process

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about ...

**0**

votes

**0**answers

21 views

### Expected finite time Queue Length in Birth-Death process

Consider a Birth-death process $Q(t)$ with reflection at zero with geometric arrival and departure. With probability $\lambda$ there is an arrival at each time-slot. At the end of each time-slot there ...