**1**

vote

**0**answers

61 views

### question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
...

**4**

votes

**2**answers

87 views

### Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$

Assume a process with Itô dynamics of the generic form
$$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$
and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
...

**1**

vote

**1**answer

95 views

### Convergence of a sequence of dependent binomial trials

Consider a sequence of the the following stochastic process.
Let $b_0=1$, and let $n >1$. At each step $t$, let $b_t \sim Bin(n,\frac{b_{t-1}}{n})$.
The process stops when either $b_t=0$ $b_t=n$. ...

**1**

vote

**0**answers

44 views

### forward algorithm Hidden Markov Model

I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that that you are trying to propagate through a sequence (and the available states) to find the most ...

**1**

vote

**0**answers

84 views

### Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function ...

**4**

votes

**2**answers

152 views

### Joint probability distribution as functions

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...

**1**

vote

**0**answers

14 views

### How can I identify correlation between event types over time [migrated]

I have many thousands of event types. Some very frequent, but most fairly infrequent. I want to identify events types that could be related by virtue of the fact that they occur relatively closely ...

**1**

vote

**1**answer

94 views

### Strong solutions on SDE (stochastic differential equations) with discontinuous drift and diffusion coefficients

I want to get some advice from you about the existence (and the uniqueness if possible) of a strong solution on my SDE. In fact, due to the structure of the problem that I consider, both the drift ...

**3**

votes

**1**answer

121 views

### conditional expectation under convex combinaison of probability measures

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 ...

**0**

votes

**1**answer

55 views

### Estimating the variance of error in empirical approximation to a distribution

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
...

**1**

vote

**1**answer

63 views

### What is known about the distribution of the errors in empirical approximation of a CDF?

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
...

**6**

votes

**0**answers

105 views

### Sampling from a Convex Body with Many Extremal Points

Let $p_{1}, \ldots, p_{N}$ be a collection of points in $\mathbb{R}^{n}$. I would like to sample uniformly from the convex hull of these $N$ points in an `efficienct' way. In my setting, I have $n$ ...

**6**

votes

**1**answer

198 views

### An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have
$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...

**2**

votes

**1**answer

77 views

### A strange definition about reproducing kernel Hilbert spaces for symmetric Gaussian measures on a separable Banach space

In stochastic equations in infinite dimensions written by Da Prato reproducing kernel Hilbert space is defined
Let $\mu$ be a symmetric Gaussian measures on a separable Banach space $E$. A linear ...

**2**

votes

**1**answer

89 views

### Positive correlation of conditional expectations

Suppose I have a random variable $Z$ defined on a probability space built on a $\sigma$-algebra $\mathcal{F}$ (with $\mathbb{E}[Z^2] < \infty$). Moreover, suppose that I have two filtrations ...

**7**

votes

**1**answer

485 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**7**

votes

**3**answers

229 views

### Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...

**0**

votes

**0**answers

38 views

### Closed for the motion of an interacting particle system

I am dealing with interacting particle systems approximately in the sense of http://www.math.vu.nl/~rmeester/onderwijs/Interacting_Particle_Systems/liggett.pdf p. 5 except I am reading a book by the ...

**2**

votes

**0**answers

66 views

### Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics.
I am already ...

**3**

votes

**1**answer

96 views

### Proof for the emergence of a ranking with paired comparisons [closed]

Take a set {A, B, C, D, E}, and assume each of the set elements has a random real value attached to it between 0 and 1. For example, this gives us: {A, B, C, D, E} = {0.1, 0.9, 0.4, 0.6, 0.5}. Assume ...

**1**

vote

**1**answer

59 views

### Measures of disjoint unions and complements of a collection of sets

Let $\mu$ be a probability measure. Let $\mathcal A$ be a collection of measurable sets and $D(\mathcal A)$ be the minimal $\lambda$-system (Dynkin system) containing $\mathcal A$.
Is $\mu(D)$ for ...

**-1**

votes

**0**answers

66 views

### conditional expectation of discrete random variable given sum constraint [migrated]

Given $k$ discrete iid random variables $X_1,\ldots,X_k$, the conditional expectation
$E[X_1\,|\,X_1+\cdots+X_k=n]$
is intuitively given by
$\frac{n}{k}$
since the variables are iid and, thus, ...

**3**

votes

**1**answer

155 views

### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...

**7**

votes

**0**answers

78 views

### Longest induced cycles in random geometric graphs near criticality

We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...

**3**

votes

**1**answer

46 views

### Random weighted selection without replacement

I am using the following procedure to select $m$ different numbers $\{i_1,\ldots,i_m\}$ from the set $\Omega = \{1,\ldots,N\}$, with $m,N\in\mathbb{N}$ such that $m< N$.
Selection procedure
...

**1**

vote

**0**answers

58 views

### asymptotic estimate of random walk involved hitting time and return time

Consider a reversible random walk on (say) $\mathbb{Z}$, are there any estimate for the following probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and ...

**19**

votes

**1**answer

989 views

### Expected halting time for “The 2^n Game” (aka 2048) — with random moves

Recently I encountered an online flash game that features an m-by-m grid and input from the directional pad (up, down, left, right). At any point in the game, the grid contains numbers ('blocks') from ...

**4**

votes

**0**answers

132 views

### # roots of polynomials over GF(2)

Consider a polynomial $p(x)$ with of degree $d$ with coefficients in $GF(2)$. How many roots can it have in $GF(2^m)$? The intent here is that $d \ll 2^m$.
The trivial bound is of course $\leq d$. ...

**2**

votes

**0**answers

34 views

### Finding the number of leaf nodes at specific level of a random tree

Given a uniform recursive tree (URT) of size $N$ rooted at one node whereby the tree is generated as follows:
Starting with a root node, at each iteration, a new node is connected to one of the ...

**2**

votes

**0**answers

63 views

### Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...

**3**

votes

**0**answers

74 views

### Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence:
$ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $
How can ...

**2**

votes

**2**answers

395 views

### Sum of two independent random variables

Let $\xi, \eta, \eta'$ be non-negative random variables such that:
$\eta \stackrel{\mathcal{L}}{=} \eta'$,
$\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$,
$\xi$ and $\eta'$ are independent.
...

**2**

votes

**1**answer

492 views

### Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...

**9**

votes

**1**answer

313 views

### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

**1**

vote

**0**answers

25 views

### Searching for the maximum of a (strictly convex) two-dimensional distribution via maximization over a series of arbitrarily specified 1D intervals

Let $f$ be a strictly convex two-dimensional distribution with a maximum $M$ at some unknown position $(x_m,y_m)$. Starting from the origin, $(x_0,y_0) = (0,0)$, we need to find $(x_m,y_m)$, however ...

**14**

votes

**2**answers

447 views

### Spearing rolling hula hoops

Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...

**13**

votes

**4**answers

451 views

### Are gaussians with different moments far in total variation distance?

If two Gaussians disagree on one moment, it seems like this should imply that they have a large variation distance--equivalently, if two Gaussians are close in variation distance it's hard for their ...

**2**

votes

**1**answer

65 views

### Karhunen-Loeve expansion for discrete-time process

Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...

**0**

votes

**0**answers

45 views

### Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...

**7**

votes

**5**answers

216 views

### Nice examples/arguments that illustrating the coupling method in probability theory

I want some examples that can best illustrate the idea/power/funny of the coupling argument in probability. I think arguments with coupling makes one think in a more probabulistic way. I have a short ...

**4**

votes

**0**answers

129 views

### Conditional expectation with respect to random closed sets

Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...

**2**

votes

**0**answers

66 views

### Link between presence of attracting random fixed points and synchronisation - is this an open question?

This is a question in the theory of random dynamical systems.
Let $(X,d)$ be a compact metric space, let $(I,\mathcal{I},\nu)$ be a probability space, and let $(f_\alpha)_{\alpha \in I}$ be an ...

**1**

vote

**0**answers

65 views

### On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth ...

**4**

votes

**1**answer

80 views

### General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...

**5**

votes

**1**answer

103 views

### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...

**4**

votes

**1**answer

131 views

### Reducing search space by probability

First question on this Stack, so I hope this is appropriate. This is not for homework, but a personal question related to a machine learning problem I am working on. However, I don't have enough ...

**0**

votes

**3**answers

193 views

### Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?

The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?

**6**

votes

**2**answers

393 views

### Probability of Brownian motion to have a zero in an interval

I have what should be a very simple questions for Brownian motion experts...
Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at ...

**2**

votes

**1**answer

123 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...

**1**

vote

**0**answers

150 views

### An extrasensory perception strategy :-)

I asked this question at MSE some months ago
but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a ...