Tagged Questions
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
5
votes
1answer
65 views
What are all the stationary and pointwise independent random processes?
In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the ...
0
votes
2answers
72 views
Determine joint distribution from projections
Let $X=(X_1,\dots,X_d)$ be a random vector, and a.s. $X \in [0,1]^d$. Suppose that for every $a \in \mathbb{R}^d$, we know the probability distribution of the random variable $Y_a = <a,X>$. My ...
2
votes
0answers
36 views
Probability that convex hull of multivariate Gaussian sample contains a given point
I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls ...
6
votes
2answers
207 views
Can every discrete martingale be embedded in a continuous martingale?
Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
1
vote
2answers
114 views
Smooth but non-analytic kernel functions
Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
2
votes
0answers
53 views
MLRP of random variables and order statistics
Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy the monotone likelihood ratio ...
-6
votes
0answers
44 views
Stop Loss insurance [on hold]
I need help in this question, I shall be very thankful if anyone can help me in it.
0
votes
0answers
77 views
Derivative of Set Functions
For any closed set $I\subseteq (0,1]$, I have a set function in the form of
$$\theta(I)=\int_{q(I)}YdF(Y)$$
Where $q(I)$ if the image of I under the mapping $u \mapsto q(u)$. And $F(\cdot)$ is the ...
1
vote
0answers
120 views
Random operators [migrated]
Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
4
votes
0answers
83 views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
6
votes
1answer
347 views
Mean of i.i.d Random Variables With No Expected Value
Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
2
votes
1answer
102 views
Proving that a closed walk of some odd length k exists on a Cayley graph
I'm trying to prove the following:
Let $G$ be a group with finite symmetric generating set $S$ and let $\Gamma(G,S)$ be the corresponding Cayley graph. Let $X_1, X_2,\cdots$ be a simple random walk ...
3
votes
1answer
107 views
General additive function of probability
Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:
$H$ is continuous,
$H$ is symmetric w.r.t. the order of its arguments,
...
14
votes
2answers
674 views
Randomly walking a leashed dog
Let a human $h(t)$ random walk on $\mathbb{Z}^2$ by taking a unit-length step at every
time step $t$. A dog $d(t)$ on a leash of length $\lambda$ follows $h(t)$, also
taking a unit-length step at ...
3
votes
1answer
82 views
Variance of maximum of mixture of gaussians
Let $\{X_i\}$ be an iid collection of standard normal $(N(0,1))$ random variables . Let $X = (X_1,\ldots,X_n)$, and consider a function of the form $f(X) = \max(A\cdot X)$, where $A$ is some ...
3
votes
0answers
41 views
Cramér-Wold theorem with independence assumption
Let $X = (X_1, X_2)$ be a random vector with joint probability density $p$. The celebrated Cramér-Wold theorem says that we can reconstruct $p$ from knowing the push-forward densities of $X$ under all ...
2
votes
1answer
53 views
right-continuity of filtration
For a natural filtration of a stochastic process (possibaly multi-dimensional) to be right-continuous, what conditions should the process satisfy? Any references?
0
votes
0answers
20 views
Multiplying pdf's: Statistics [migrated]
If I have two different independent continuous variables, X and Y. Is it legitimate to multiply their pdf's to form a pdf for the ordered pairs (X,Y) as f(x,y).
I've plotted the surface and the ...
11
votes
1answer
158 views
Wander distance of self-avoiding walk that backs out of culs-de-sac
Suppose a self-avoiding walk on $\mathbb{Z}^2$, with random steps each one unit long,
backs out of culs-de-sac, but retaining the lattice points on which it stepped
marked as unavailable for future ...
-1
votes
0answers
23 views
Inference in undirected graphical models. [migrated]
I know Bayesian or Directed Graphical models are good for fast inference using message passing techniques. But how does it work with undirected models? With directed models, you moralize the graph ...
2
votes
0answers
53 views
Are there pathological examples of log-concave measures that admit no shifts?
Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties?
The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
1
vote
2answers
138 views
Methods of probability theory in differential geometry fruitful? [closed]
I am trying to understand how the two paradigms of differential geometry and probability theory can fruitfully be applied to each other.
The more suggestive direction is to use methods of ...
5
votes
1answer
167 views
1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
...
-1
votes
0answers
18 views
Probability Analysis of a Randomized Algorithm [migrated]
A rock band has three sites A, B, and C that it needs to perform at. The band performs at site A, then randomly chooses between B and C as to where it performs next. The band keeps choosing one of the ...
1
vote
0answers
54 views
asymptotic variance of sample autocorrelation of two iid random variables
I am trying to prove that the variance of the sample lag-1 autocorrelation
$$\hat{\rho}=\frac{\sum_{t=1}^n(x_t-\bar{x})(x_{t-1}-\bar{x})}{\sum_{t=1}^n(x_{t-1}-\bar{x})^2}$$
for an i.i.d. R.V is ...
4
votes
2answers
505 views
Do Random Walks on the Hexagonal Lattice have a limit?
For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of ...
-3
votes
0answers
14 views
Moment of uniform distribution [migrated]
Suppose that $U$ is a random variable from a uniform distribution on $[a, b]$.
Then, we can obtain the moment generating function of $U$, and by using that, we can get the $n$th order moment of $U$ ...
9
votes
0answers
154 views
What kind of random matrices have rapidly decaying singular values?
I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...
1
vote
1answer
58 views
The jump and the left martingale of semimartingale
Let $X_{t}$ be a semimartingale. Define
$\Delta X_{t} = X_{t}- X_{t-}$.
For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...
5
votes
2answers
134 views
If Gaussian measures on a Hilbert space converge weakly to 0, how do their covariance operators converge?
Suppose we have a sequence of Gaussian measures $N(0, S(n))$ supported on a Hilbert space $H$ and we know that the sequence converges weakly to the delta measure at $0$, what are the necessary and ...
4
votes
0answers
134 views
Does the concept of connective constant make sense for any tiling of the plane?
First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
3
votes
1answer
84 views
The regularity of Levy process
There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$.
This property is called the ...
12
votes
1answer
325 views
Probability that random cubic polynomials meet in a square
Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...
1
vote
0answers
104 views
Galton/Plinko Board Manipulations - Theoretical Question [migrated]
I am familiar with a "Galton Board"/Plinko board basically illustrating the central limit theorem. What would happen if:
I changed the distribution of pegs on the board?
I changed the "incoming ...
0
votes
0answers
181 views
Sum over a product of binomial coefficients related to a collision problem
I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:
...
4
votes
1answer
74 views
Connection between degree of growth and return probabilities of random walks on Lie groups
Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...
0
votes
1answer
117 views
First order stochastic dominance and rearrangement inequalities
I have N independent random variables $X_1, \cdots , X_N$ all of which are positive. Let $X_i \sim F_i$ where $F_i$ is the CDF of $X_i$.
Also, $F_N > F_{N-1} > \cdots > F_1$ where $>$ ...
4
votes
0answers
119 views
Maximum distance between two consecutive points of N random points on a unit length line
I have encountered a seemingly simple question on distances of random points.
Place N points randomly and uniformly on the line segment [0..1].
How to derive the expectation (or the distribution) of ...
3
votes
2answers
172 views
Nonmonotonicity of expected distance of a random walk
What's the simplest example of a reversible random walk $X_n$ on an infinite vertex-transitive graph such that the expected distance from the origin is not increasing, i.e. there exists $n$ such that:
...
1
vote
1answer
66 views
Stokes-Einstein rotational diffusion and vector orientation time
The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as:
$D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
2
votes
0answers
37 views
Families of continuous random variables closed under multiplication and taking maximum
I am looking for a finitely parameterized family of non-atomic distributions $D(\vec{\lambda})$, $\vec{\lambda} \in \mathbb{R}^k$ for some finite $k$, such that if $X \sim D(\vec{\lambda}_1)$ and $Y ...
3
votes
0answers
103 views
Tight lower bound for expected maximum of K sums of T Rademacher random variables
For each $j \in \{1, \ldots, K\}$, let $(\varepsilon_{j,t})_{t=1}^T$ be an independent sequence of iid Rademacher random variables (i.e. taking values $\pm 1$ with equal probability). What is the best ...
3
votes
0answers
48 views
The distribution of Jump gaps of Levy process
Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
5
votes
1answer
152 views
Convergence rate of the central limit theorem near the center of the distribution
I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...
5
votes
3answers
275 views
Deconvolution of sum of two random variables
Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$
I have observations of $Z_i$'s and thus can ...
1
vote
1answer
63 views
explicit expressions of the distribution of sums of i.i.d. logistic random variables
Where can I find the explicit expression of the distribution of the sum of n i.i.d. logistic random variables, for n=2,3,4...
The expressions given in "On the convolution of logistic random ...
4
votes
0answers
184 views
Inverse of matrix-valued function
Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by
\begin{equation}
...
1
vote
0answers
48 views
What is entropy of a variable described by Knightian uncertainty? [closed]
I have asked this question at Theoretical Computer Science and received no response.
Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, ...
4
votes
1answer
210 views
Probability distribution or the distance between two points in $n$-dimensional Euclidean space after a random perturbation of one point
Take two points, $p_0$ and $p_k$, in $n$-dimensional Euclidean space, where $d(p_0,p_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p_0$ and uniformly ...
2
votes
1answer
250 views
How fast does the Heat equation with boundary condition $\frac{\partial u}{\partial \vec{n}}=u^2$ decay?
Consider the heat equation $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u$ in a bounded domain (say the interval [0,$\pi$]) with boundary condition $$\frac{\partial u}{\partial \vec{n}}=u^2$$
with ...
