Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
2
votes
0answers
35 views
Almost-Monotone Kernels - Examples and/or Covering Theorems
I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation.
Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures ...
6
votes
2answers
122 views
Random walk in a convex body or convex polytope
Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in
$\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside ...
5
votes
0answers
80 views
Concentration of sum of powers of normals
Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...
0
votes
1answer
80 views
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)
How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
2
votes
1answer
105 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 ...
4
votes
1answer
71 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
1answer
39 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
0answers
86 views
Probability of differently loaded dice summing to a value [on hold]
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
1answer
113 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
0answers
81 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
1answer
65 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
1answer
148 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 ...
1
vote
0answers
185 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
1answer
119 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
0answers
81 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
0answers
39 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
1answer
181 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
2answers
91 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
2answers
85 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
1answer
85 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
0answers
76 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
1answer
158 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
0answers
138 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
1answer
209 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
0answers
61 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
0answers
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
1answer
81 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
0answers
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
0answers
76 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
2answers
204 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 ...
1
vote
0answers
34 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 $\lambda$. ...
0
votes
0answers
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 ...
1
vote
1answer
73 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
0answers
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
0answers
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
1answer
78 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
1answer
139 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
1answer
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
1answer
59 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
1answer
66 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
0answers
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
1answer
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
0answers
120 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:
...
3
votes
1answer
110 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
0answers
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
0answers
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
2answers
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
0answers
83 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
2answers
120 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
1answer
46 views
Finiteness of “novel variance” from a kernel on a compact space [closed]
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, ...
