**1**

vote

**1**answer

104 views

### Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...

**0**

votes

**0**answers

35 views

### Probability distribution/ Statistics of noise Eigenvectors

What is the distribution of noise Eigenvectors of the sampled covariance matrix? For the signal Eigenvectors the expressions are well defined which approach infinity in the noise Eigenvectors case; ...

**2**

votes

**1**answer

166 views

### Asymptotic behavior of a ratio of sums of iid random variables

Let $X_i$ and $Y_i$ be distributed identically to $X$ and $Y$, respectively. Assume both $X$ and $Y$ take strictly positive values.
Consider the random variable $R_n \doteq \frac{\sum_{i=1}^n X_i}{\...

**2**

votes

**0**answers

95 views

### On the Bhattacharyya distance

Let $X$ and $Y$ be two continuous random variables with support $\mathbb{R}^{+}$ and with PDF $f(x)$ and $g(y)$. If the Bhattacharyya distance of $f$ and $g$ is less than $\epsilon$, then is there any ...

**1**

vote

**0**answers

55 views

### A variance-preserving Boolean function [closed]

Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that $\mathsf{var}(...

**-1**

votes

**1**answer

155 views

### Regarding a new divergence function of two probability distributions

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any $0 \leq \alpha \leq 1$, and any constant $\beta$ within the support of $X$ and $Y$ such ...

**0**

votes

**0**answers

81 views

### Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where $\mathsf{prime}$...

**4**

votes

**1**answer

189 views

### How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...

**0**

votes

**0**answers

29 views

### Bayesian model to estimate the parameter of a Bernoulli law

Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ...

**0**

votes

**1**answer

332 views

### What is the derivative of this integral?

I have asked this question here
http://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...

**1**

vote

**0**answers

66 views

### Monte Carlo Simulation - efficient simulation of tail outcomes [closed]

When running Monte Carlo type simulations in situations where you're only interested in tail outcomes, do you know of a way to only simulate those outcomes, so that you can come up with more reliable ...

**2**

votes

**1**answer

40 views

### Conditioned binomial dominates unconditioned with different parameter

Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show
$$(X\mid X \geq 1) \succeq_{sd} 1 + Y.$$
Here $(X \mid \cdot)$ is the ...

**1**

vote

**1**answer

143 views

### Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...

**3**

votes

**1**answer

160 views

### density of distance between points in unit circles

Let $a$ and $b$ be two points in the plane. Let's choose a point $c$ uniformly from the circle of radius $r$ with $a$ as center and choose a point $d$ uniformly from the circle of radius $r$ with $b$ ...

**10**

votes

**1**answer

171 views

### Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...

**1**

vote

**1**answer

148 views

### moment sequence which does not define a random variable vs convergence in distribution

I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X_n\}_{n\geq 0}$ from which we know ...

**1**

vote

**1**answer

87 views

### Exponentially Bounded Sequence of Moments defining Distribution?

I have an exponentially bounded sequence $m_n = \lambda^n + c_n$ (i.e. the $c_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying ...

**0**

votes

**0**answers

42 views

### Convergence of generalized inverses

During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation to post more links), I've got stuck, trying to understand more deeply one of ...

**2**

votes

**2**answers

135 views

### Convergence in distribution to a Poisson

We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity.
The ...

**10**

votes

**2**answers

227 views

### Slight variation on law of the iterated logarithm

Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\...

**3**

votes

**1**answer

90 views

### Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e.
$$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} p_1^{n_1}...

**4**

votes

**1**answer

78 views

### What class of probability distributions do probabilistic turing machines induce? [closed]

What class of probability distributions is induced by the class of probabilistic turing machines? Is there a precise characterization?

**0**

votes

**0**answers

49 views

### methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...

**10**

votes

**1**answer

304 views

### Extension of Dynkin's formula, conclude that process is a martingale

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...

**4**

votes

**0**answers

79 views

### Concluding that the Poisson kernel is indeed the Cauchy distribution?

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...

**3**

votes

**1**answer

139 views

### Poisson kernel, expectation, an absolute value comes in

See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...

**2**

votes

**1**answer

123 views

### Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...

**8**

votes

**2**answers

275 views

### Distribution of $\max_{n \ge 0} S_n$, random walk

Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...

**4**

votes

**1**answer

119 views

### Is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane almost surely?

Let $d = 2$. With probability $1$, is the set of multiple points of the Brownian path $W[0, \infty)$ dense in the plane?

**1**

vote

**1**answer

191 views

### Formula for maximum of two Gumbel distributions?

I have parameters of two Gumbel distributions ($\mu_1, \beta_1)$ and $(\mu_2, \beta_2)$. Since max of 2 Gumbels is a Gumbel, I'd like to compute $\mu_m, \beta_m$, so that:
$Gumbel(\mu_m,\beta_m)$ = $...

**2**

votes

**1**answer

132 views

### Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?

**1**

vote

**0**answers

29 views

### Importance sampling for bernoulli-sequence, favouring long sequences of ones

Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$.
I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...

**7**

votes

**1**answer

201 views

### Brownian motion, crossing intervals, possible usage of second moment method?

This is a followup to my question here.
Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$...

**11**

votes

**4**answers

431 views

### Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...

**0**

votes

**0**answers

47 views

### Order statistics of the diagonal terms of an inverse Wishart matrix

I have a question about the inverse Wishart matrix. In my understanding, consider $\mathbf H$ is a $n\times n$ matrix with each elements are complex Gaussian with zero mean unit variance. Then $\...

**8**

votes

**1**answer

226 views

### If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$?

Suppose that we have distributions $F_1 $ and $F_2$. Under what conditions on $F_1,F_2$ is it possible to construct random variables $X\sim F_1,Y\sim F_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ ...

**6**

votes

**2**answers

124 views

### For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large $M_n \le r\sqrt{\log n}$?

Let $B_t$ be a standard Brownian motion. Let$$M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\}.$$For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large$$M_n \le r\...

**0**

votes

**1**answer

140 views

### Discrete random walk with uniformly distributed transition p, set initially

I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:
Choose $p$ from $U(0,1)$
Start ...

**1**

vote

**2**answers

128 views

### Non-normality of limit of random variables

I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that
$$\mathbb{E}[X_n]=\...

**2**

votes

**0**answers

74 views

### Construct a sequence of i.i.d random variables with a given distribution function, diagonalization? [closed]

Assume we have a sequence of i.i.d. random variables $X_1, X_2, \dots,$ on a probability space $(\Omega, \mathcal{F}, P)$ with$$P(X_n = 1) = P(X_n = -1) = {1\over2}.$$Given a distribution function $F$,...

**3**

votes

**1**answer

134 views

### Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu_n$, the distributions of $...

**5**

votes

**0**answers

75 views

### Joint cumulants of $Z_2^n$ characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...

**10**

votes

**1**answer

286 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**3**

votes

**2**answers

152 views

### splitting exponential random variable into independent components

$X$ follows Exponential $(\lambda)$. Can we split $X$ into two independent r.v.'s, i.e.,
do there exist functions $g$ and $h$ such that $g(X)$ and $h(X)$ are independent for any fixed $\lambda$? $g(X)...

**7**

votes

**2**answers

221 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant $K>0$...

**3**

votes

**1**answer

98 views

### Reference request for a result regarding density of induced probability measure under a submersion

Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map ...

**6**

votes

**1**answer

139 views

### Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$

Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That said,...

**0**

votes

**0**answers

66 views

### Is this probability distribution studied in literature?

Let $\theta_1,\theta_2,\theta_3$ be 3 non-negative random variables such that $\theta_1+\theta_2+\theta_3=1$ with the joint probability distribution
\begin{align}
P(\theta_i)\,=\,K\frac{\theta_1^{a_1}...

**2**

votes

**0**answers

53 views

### What is the Blumenthal-Getoor index of Student's distributions?

For infinitely divisible random variables, Blumenthal and Getoor introduced in [1] an index that allow to study for instance the local Hölder regularity of Lévy processes. For an infinitely divisible ...

**1**

vote

**2**answers

321 views

### The Levy measure of the compound Poisson distribution

The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...