Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

learn more… | top users | synonyms (1)

0
votes
0answers
15 views

Bounding Expected Value of a piecewise function

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
1
vote
0answers
71 views

Distribution of a transformed Gaussian random vector

Let $B$ be a random vector with $b_{i} \backsim N(m_i,\sigma_i) $ and $Y$ another random vector with $y_i \backsim N(r_i,\psi_i)$. Let $A$ be a symmetric and non-singular square matrix. What is ...
2
votes
0answers
34 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
0answers
105 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
0answers
73 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
1answer
72 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.
3
votes
1answer
140 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...
2
votes
1answer
123 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 ...
-1
votes
0answers
19 views

Statistics, the deviation and expection of a number sequence [on hold]

There is a sequence of number $a_{0},a_{1},...,a_{n}$, $(0 < a_{i} < 1)$ Define $b_{t} = \frac{ \sum_{i=0}^{t}{w^{t-i}a_{i}} }{ \sum_{i=0}^{t}{w^{t-i}} }$ where $w \in (0, 1)$. Can we proof ...
0
votes
0answers
56 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
0answers
57 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 ...
-1
votes
0answers
53 views

Probability and Statistics [on hold]

for the cards shown below, what is the probability of choosing a yellow card and then a D if the first card is replaced before the second card is drawn? [b] [1] [5] [D] [10] ...
3
votes
0answers
96 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
0answers
64 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
1answer
94 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 ...
9
votes
2answers
620 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
0answers
35 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 ...
-2
votes
0answers
33 views

Having the highest value in a interval appear less often [closed]

I have an array of size 5. And initially in each index, they are initialized with the value 1. so it looks like this : 1 1 1 1 1 Every iteration, I get a decimal value between 0.0 and 1.0 At the ...
10
votes
5answers
434 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
0answers
39 views

Convergence of approximate quadratic variation in $L^p$

For a semimartingale $X_t$, 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 ...
0
votes
0answers
23 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
0answers
47 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
0answers
80 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
0answers
235 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
0answers
71 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
1answer
158 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
2answers
107 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
1answer
198 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, ...
0
votes
0answers
27 views

kernel and operator of determinantal point process

is it true that that when the space is discrete & finite ($X=\{1,2,\ldots,n\}$) the kernel of determinantal point process and operator of it are the same?
1
vote
0answers
95 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
2answers
137 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
0answers
20 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
1answer
71 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
0answers
30 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
1answer
142 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
0answers
40 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
1answer
146 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
1answer
50 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
0answers
46 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
0answers
39 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
1answer
21 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
2answers
80 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
0answers
93 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
0answers
77 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
2answers
97 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
1answer
130 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 ...
5
votes
2answers
199 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
0answers
17 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 ...
1
vote
0answers
34 views

Basic results for chi square processes

I could not find any introductory material with basic results regarding chi-square processes. Their definition from The Supremum of Chi-Square Processes is as a sum of $d$ squares of independent ...
6
votes
1answer
243 views

Forbidden coin flips

Suppose I have a (possibly infinite) bag of coins with various weights. I select a coin and flip it $n$ times. Averaging over the choice of coins from the bag, there is some probability of seeing ...