Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
1
vote
0answers
94 views
A complicated combinatoric/probabilistic limit
I'm hoping to find a tractable expression for a limit of the following expression:
$P_i (\textbf{n},\textbf{U};M,N) =\frac{M U_i}{M^{\sum \limits_{j=1}^{N} n_j}} \sum \limits_{k_1 = \delta_{i,1}} ...
3
votes
0answers
59 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 ...
0
votes
0answers
38 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
32 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
168 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
78 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
80 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 ...
2
votes
1answer
79 views
M/M/1 Queue with probability of new customer leaving
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 = ...
1
vote
0answers
61 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
141 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
123 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
202 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
59 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
103 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
72 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
43 views
vector-matrix notation and expectation of matrix and Hermitian product [on hold]
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
70 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
186 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
0answers
30 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
0answers
60 views
Books and papers on differential equation method [on hold]
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
0answers
76 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
0answers
42 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
44 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
35 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
74 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
136 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
73 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
55 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
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
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
53 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
111 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
1answer
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 ...
-1
votes
0answers
42 views
Feature relationship based class separability [closed]
I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.
I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
-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
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, ...
9
votes
2answers
111 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
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
0answers
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
1answer
644 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 ...
-1
votes
2answers
99 views
conditional expectation under convex combinaison of probability measures(II)
Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...
1
vote
3answers
105 views
Empirical estimator for total variation distance between two product distributions
Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
5
votes
1answer
162 views
Measures which exhibit the “uncorrelated implies independent” property
Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...
0
votes
0answers
68 views
Blackwell-MacQueen Urn Scheme
I'm having some difficulty understanding the steps in the proof in the bottom half of the second page of this paper: ...
5
votes
3answers
419 views
Does $Mv$ converge to i.i.d in some sense?
I am not a professional mathematician so please excuse me if my question is not phrased correctly.
I am interested in the following simple sounding problem.
Consider a random $n$ by $n$ $0$-$1$ ...
0
votes
0answers
45 views
Hermite coefficients of a positive density
It is well known that a necessary condition for a function in $L_2$ to be a.e. positive is that its fourier transform is positive-definite (in fact, due to Bochner's theorem, this is also a sufficient ...
0
votes
1answer
57 views
Cramér-Wold device with limited angle and independence assumption
Let $X$ be a random vector taking values in $\mathbb R^2$ with probability density $p(x) = p_1(x_1)p_2(x_2)$, i.e. the components of $X$ are independent.
Let $V$ be an open set in $\mathbb S^1$, the ...
2
votes
0answers
115 views
Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
