**0**

votes

**0**answers

5 views

### Parameter estimation using bayesian update on moduli space?

Scientists take a set of data points, say in ${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree $d$ (or an exponential, etc.), they estimate parameters.
I would think ...

**3**

votes

**1**answer

43 views

### On the moments of Lévy processes

For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule:
$$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$
with $\kappa_p<\infty$. The proof only requires to remark ...

**2**

votes

**1**answer

32 views

### Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Model:
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rates $\lambda$ and service rates $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) discipline ...

**0**

votes

**0**answers

71 views

### Probability of differently loaded dice summing to a value

I have a real world problem that boils down to the following:
I'm playing dice. I have $n \approx o(10)$ differently biased die. The probability of the $i^{th}$ die showing $x_i$ is given by ...

**2**

votes

**0**answers

67 views

### Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...

**1**

vote

**0**answers

67 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h ...

**-1**

votes

**1**answer

64 views

### Clique factorization

I'm reading about Clique factorization in wikipedia:
http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization
but I'm unable to understand this:
What is $X_C$ here? Ok I understood ...

**6**

votes

**1**answer

126 views

### Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral
$$ X_T:= \int_0^T W_t dt $$
It is often mentionend in literature that $X_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am ...

**2**

votes

**0**answers

132 views

### bound for $|E[\frac{X}{Y}]−\frac{E[X]}{E[Y]}|$

Is there some bound for $|E[X/Y]−E[X]/E[Y]|$ ? Here $X$ and $Y$ are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is to guarantee that the denominator ...

**1**

vote

**0**answers

183 views

### A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version:
Randomly assign (with replacement) $N$ balls to $M$ urns. ...

**4**

votes

**1**answer

116 views

### What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes,
Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards ...

**1**

vote

**0**answers

76 views

### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the
Fourier transform of a random variable $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**1**

vote

**0**answers

37 views

### A counterpart of Karhunen theorem

According to the Karhunen theorem, if the correlation function of a process $X(t)$
can be represented as
$$
R(t,s)= \int_{\Lambda} f(t, \lambda) \overline{f(s, \lambda)}d\nu(\lambda)
$$
then the ...

**2**

votes

**1**answer

179 views

### What is “tilting” in the context of large deviations?

I have seen references to the "tilting method" in the theory of large deviations. Is there a simple explanation of what this is, exactly?

**2**

votes

**2**answers

87 views

### Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.
1) Does this quantity $f(X,t)$ have a name? ...

**1**

vote

**2**answers

84 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where ...

**1**

vote

**1**answer

83 views

### M/M/1 Queue with probability of new customer leaving [on hold]

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...

**2**

votes

**0**answers

73 views

### Finite Volume 1D Anderson Tight Binding Model

My question is about bounds on the number of eigenvalues in a microscopic interval for the random Schrodinger operator on $\mathbb{Z}_n$ for $n \in \mathbb{N}$. For my question, these are the ...

**6**

votes

**1**answer

154 views

### Limit of distance between two random points in a unit-radius $n$-sphere

This is a companion contrast to the earlier analogous question for unit $n$-cubes,
where the answer (provided by several respondents) is $\infty$ .
What is the limit, as $n \to \infty$, of the ...

**0**

votes

**0**answers

129 views

### Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...

**2**

votes

**1**answer

205 views

### How to minimize $-\sum p_b \ln{p_b}$?

Consider multisets of the form $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of ...

**1**

vote

**0**answers

60 views

### On Flajolet's analytic urn model: a unified approach or just an interesting trick?

Recently I'm reading Flajolet's work on analytic urn models. In around 2006 He introduced a new analytical method that can give exact solutions to many classical urn models in a unified way. For a ...

**2**

votes

**0**answers

113 views

### Heat kernel and Wiener measure

A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...

**0**

votes

**1**answer

79 views

### Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...

**-1**

votes

**0**answers

44 views

### vector-matrix notation and expectation of matrix and Hermitian product [closed]

Let $\textbf{h} \in \mathbb{C}^{N\times 1}$, $\textbf{a} \in \mathbb{C}^{N\times 1}$, $\textbf{b} \in \mathbb{C}^{N\times 1}$ and $\textbf{c} \in \mathbb{C}^{N\times 1}$. The variable $h_i$ is defined ...

**1**

vote

**0**answers

74 views

### number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...

**6**

votes

**2**answers

200 views

### Limit of distance between two random points in a unit $n$-cube

What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average ...

**0**

votes

**0**answers

32 views

### Bounding Rayleigh quotioent for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...

**-2**

votes

**0**answers

61 views

### Books and papers on differential equation method [closed]

I wanted to understand the differential equations method for analyzing stochastic sequences. Is there a good book/ papers that provide a gentle survey this topic with a good number of examples? A good ...

**0**

votes

**0**answers

86 views

### Upper bound for $r_{0}(n)$ through probabilities

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are primes. For a given $N$, let's denote by $r_{0}(N)$ the ...

**0**

votes

**0**answers

43 views

### Quantiles moments and Convergence

QUESTION:
Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence
...

**1**

vote

**0**answers

46 views

### Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...

**0**

votes

**0**answers

37 views

### Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian?
In ...

**2**

votes

**1**answer

77 views

### convergence rate of occupation measure of ergodic Markov Chain

Given an ergodic Markov chain $(X_n)_{n\geq 1}$ in $R^d$with $\pi$ as the invariant distribution of the transition kernel, under good conditions we have that the empirical occupation measure converges ...

**1**

vote

**1**answer

137 views

### Question about the log-det function

Suppose I have a diagonal $n \times n$ matrix $\Gamma$ with positive entries, and a fixed $n \times k$ matrix $P$ with $P^\intercal P = I$ (here, $k \leq n$). I'm interested in knowing whether the ...

**2**

votes

**1**answer

74 views

### Convergence of random variables in LP preserved under conditioning on sub sigma field

Is anyone aware of a result which states that convergence of random variables in $\mathbb L^p$ are preserved under conditioning on sub-sigma fields?
I'm new to probability/measure theory, and trying ...

**1**

vote

**1**answer

58 views

### A calculation involving a uniform random variable quantile

THE PROBLEM:
Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to
...

**1**

vote

**1**answer

65 views

### question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
...

**2**

votes

**0**answers

109 views

### Equivalence of Gaussian measures on Hilbert space

Suppose we have 2 nondegenerate Gaussian measures given by N(0,T) and N(0,S) supported on a separable Hilbert space H. T and S are such that eigenbasis of S lies in the cameron martin space of ...

**0**

votes

**1**answer

55 views

### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...

**4**

votes

**0**answers

116 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
...

**2**

votes

**1**answer

108 views

### Calculating the probability that all possible length $r$ subwords exists in a string, with or without overlaps allowed

Let $S$ be a length $L$ string, where each character in the string is chosen with uniform random probability over an alphabet with $q$ characters. For example, a binary string would imply $q = 2$, a ...

**-2**

votes

**0**answers

12 views

### Probability of ongoing experiment [migrated]

Suppose, I do a experiment where I have an event 'a' true 1000 times in 1000 trials. So, the probability becomes 1000/1000 = 1.
If I am going to do another trial, my prediction about event 'a's ...

**-2**

votes

**0**answers

26 views

### Expectation of O_p(1) process [migrated]

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$.
Is there any condition(s) to guarantee that ...

**2**

votes

**2**answers

159 views

### Intuition on Lindeberg condition

I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?

**1**

vote

**0**answers

82 views

### mabinogion sheep problem: optimal policy [closed]

This problem occurred in Williams's book : probability with martingales, it states as follows:( from wiki )
At time t = 0 there is a herd of sheep each of which is black or white. At each time t = 1, ...

**10**

votes

**2**answers

118 views

### Intrinsic significance of differential entropy

Many commentators (e.g. Jaynes, Rota) argue that the notion of "differential entropy" is problematic (as commonly defined by $ h(X) = \int ( \log\frac{1}{p(x)} ) p(x) \, dx $, where $X$ is a random ...

**0**

votes

**1**answer

44 views

### Finiteness of “novel variance” from a kernel on a compact space [on hold]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...

**-1**

votes

**0**answers

151 views

### A question on limit of weak-* convergence of probability measures [migrated]

Let $(X,\mu)$ be a measure space. Assume $X$ is compact. It is well-known that the space $\mathcal{P}(X)$ of probability measures on $X$ is compact in weak-* topology. Let's consider a sequence of ...

**14**

votes

**1**answer

647 views

### Probability all inner products are zero

For an even (and large) positive integer $n$, consider a random $(2n-1)$-dimensional vector $v$ where each $v_i$ is $-1$ with probability $1/2$ and $1$ with probability $1/2$ and are i.i.d. Now ...