**0**

votes

**1**answer

128 views

### Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying
$$\int_{\mathbb R}xd\...

**2**

votes

**1**answer

123 views

### About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{...

**1**

vote

**1**answer

150 views

### Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression:
$$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor 1/...

**0**

votes

**0**answers

175 views

### A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration
$\hat{c} = f(c)$, where each element of $\hat{c}$ is given by
$$\hat{c}_{\ell} = \frac{\sum_k Y_{k,\ell}\frac{1}{|c_{\ell}|^2+|c_{k}|...

**1**

vote

**1**answer

130 views

### Neat definition of Harris Ergodicity

I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined.
a) What would be exactly the definition?
b) What reference could be helpful?
...

**1**

vote

**0**answers

53 views

### variance of log of ratio of chi-square variables

Let X be a chi-square variable with two degrees of freedom.
Let A and B be to arbitrary constants, with $A>B>0$.
I need the variance of
$Y=\log(1+AX)-\log(1+BX).$
The mean is, maybe not simple,...

**3**

votes

**0**answers

64 views

### Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...

**0**

votes

**0**answers

37 views

### Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...

**1**

vote

**1**answer

194 views

### connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...

**0**

votes

**0**answers

28 views

### derivation of a gap related to extreme value theory

I have an expression to evaluate as follow:
$\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$
where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows:
\...

**3**

votes

**1**answer

442 views

### Bounds on $\int \log(1+x) g(x) \mathrm{d}x$?

Let $X$ and $Y$ be two continuous real random variables with common support $(0,x_{\max}]$ and with PDF $f_X(x)$ and $f_Y(y)$. Assume that $\Pr [Y\geq\beta \mid X<\beta] \leq k$ and that $\Pr [Y<...

**0**

votes

**0**answers

22 views

### How would I derive the Hellinger distance from the Hellinger integral?

What function would I integrate to derive the following expression: $d(H_1,H_2) = \sqrt{1 - \frac{1}{\sqrt{\bar{H_1} \bar{H_2} N^2}} \sum_I \sqrt{H_1(I) \cdot H_2(I)}}\\$
I understand that this is ...

**6**

votes

**1**answer

278 views

### Functional limit theorem under random change of time

FINAL EDIT: There is one main question left: According to the answer, we have choosen $\theta=1$ , where we could choose $0<\theta<\infty$ as we like. His this sufficient, if we regarde the ...

**3**

votes

**1**answer

183 views

### Solving recursion / finding generating function of a probability mass function

I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f_n$ with the following recurrence:
$$
f_{...

**2**

votes

**2**answers

124 views

### Difference between maxima of random variables

Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x_1$, $x_2$, $y_1$, $y_2$, consider
\begin{equation}
u \equiv \max(x_1+C\, y_1, x_2+C \, ...

**3**

votes

**1**answer

138 views

### Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...

**3**

votes

**1**answer

108 views

### Learn a distribution from distributions on samples

There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...

**4**

votes

**0**answers

54 views

### Spectrum of sum of fixed matrices with random signs

Let $A_1,\ldots,A_k$ be a given sequence of $N$-by-$N$ Hermitian matrices. Assume all have spectrum contained in $[-1,-\delta] \cup [+\delta,+1]$ for some $\delta>0$. Let $$A=\frac{1}{\sqrt{k}} \...

**0**

votes

**0**answers

37 views

### A functional's expectation using both known and unknown pdf

Suppose we have a random variable $X$ with a known distribution $f$ over an interval $[a,b]$ and another r.v $Y$ over the same interval but with an unknown distribution $g$. We also have a functional $...

**1**

vote

**0**answers

48 views

### Stochastic Ordering of Negative Binomial-like Distributions

Please forgive me if this is not precise enough to post here. Simply ask me to remove it if it is not suitable. I am new here.
I am bounding the running time of an algorithm as a random variable $X$ ...

**3**

votes

**0**answers

98 views

### How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...

**8**

votes

**1**answer

286 views

### Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...

**2**

votes

**1**answer

121 views

### A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in $...

**0**

votes

**0**answers

71 views

### Proof of variance in wishart distribution

I wanna to prove the variance of wishart distribution, first a brief description of wishart distribution, how can i proof it?
I wrote a solution but the result is not correct, please help me to fix it....

**3**

votes

**1**answer

97 views

### expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution.
Background
The Wishart distribution with $\nu$ degrees of freedom and positive definite ...

**-2**

votes

**1**answer

34 views

### how to resolve the infinite nesting of interactive POMDP

I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers.
The belief update of the algorithm has a problem that agents' belief updates mutually depend ...

**4**

votes

**0**answers

183 views

### universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...

**0**

votes

**0**answers

31 views

### Hypergeometric distribution with a priori probabilities of the balls

If we have an urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then the probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...

**0**

votes

**0**answers

38 views

### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...

**1**

vote

**1**answer

277 views

### Supremum of a martingale

Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = \...

**6**

votes

**2**answers

185 views

### Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...

**5**

votes

**2**answers

155 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...

**1**

vote

**1**answer

96 views

### Distribution of maximum unique number of several random numbers

Suppose discrete random variables $\{X_1, X_2, ..., X_n\}$ are i.i.d. described by the probability function:
$f(x) \equiv \text{Pr}(X_i = x)$,
and $X_i \in \{1,2,3, ..., m\}$.
Let $Y$ be the ...

**4**

votes

**1**answer

87 views

### Question about the weak convergence of probability

Let $\mu$ be a probability measure on $\mathbb R$ and set
$$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$
Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t.
$$\int_{\mathbb R}\...

**1**

vote

**1**answer

117 views

### Question abouth Skorokhod representation of random variables (II)

This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ \...

**5**

votes

**1**answer

118 views

### Question abouth Prokhorov metric

Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that
$$E\left[|X-Y|\right]<\varepsilon.$$
Set $Law(X)=\mu$ and $Law(Y)=\nu$, ...

**0**

votes

**2**answers

131 views

### divisibility of uniform distribution [closed]

Let $X$ and $Y$ be independent and identically distributed random variables.
Can $X+Y$ be a uniform distribution?
(Please prove.)
In other words, is a uniform distribution divisible?
The meaning of "...

**3**

votes

**1**answer

157 views

### Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...

**0**

votes

**0**answers

67 views

### $\sum_{n\in\mathbb{Z}^2}d\mu(x-2\pi n)=0\Rightarrow$ the summands are pairwise mutually singular

Let $\mu$ a finite measure supported by $\Gamma$ (smooth curve in $\mathbb{R}^2$) and absolutely continuos with respect to the arc length measure on $\Gamma$.
Please why if $\sum_{n\in\mathbb{Z}^2}d\...

**1**

vote

**0**answers

37 views

### steady state of a continuous-time birth-death process

we consider a continuous-time birth-death process $\{X(t),t\geq 0\}$ with discrete state space taking non0negative integer values $\{0,1,2,3,...\}$. The transition rates of the process $\{X(t)\}$ are ...

**1**

vote

**0**answers

22 views

### distribution on arriving distinations within certain time [closed]

a traveller is travelling on a map. arriving every vertex of the map, the traveller could choose to go to next vertex according to a constant probability. The probabilities are represented in a matrix ...

**2**

votes

**0**answers

113 views

### Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...

**6**

votes

**2**answers

352 views

### A measure of how “spread out” a probability measure is

Consider a random variable $X$ whose variance is large. As a contrast to Markov's or Chebyshev's inequality, both of which measure the concentration of a probability distribution, is there a measure ...

**5**

votes

**1**answer

174 views

### Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...

**0**

votes

**0**answers

36 views

### Stochastic dominance for subsets

The subsets of a set $N=\{1,2,\ldots,n\}$ form a lattice, with larger sets being higher up, and a subset $B$ connected to another subset $A=B\cup\{x\}$ (for any $x\not\in B$) higher up by a "pipe". ...

**2**

votes

**1**answer

186 views

### Topologies for which the ensemble of probability measures is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess.
...

**1**

vote

**0**answers

42 views

### Marginal of mean from product of student-t and gamma

Let's say we have a distribution with PDF described by the product of Gamma and Student-t distributions. This is equivalent to a generative model, in which precision is first drawn from Gamma, and the ...

**2**

votes

**1**answer

100 views

### Variance of the normal CDF [closed]

Several threads (e.g. Integration of the product of pdf & cdf of normal distribution ) have shown that
$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.
I'd like to compute ...

**0**

votes

**0**answers

51 views

### Majorization in distributions of subsets

The subsets of a set $N=\{1,2,\ldots,n\}$ form a lattice, with larger sets being higher up, and a subset $B$ connected to another subset $A=B\cup\{x\}$ (for any $x\not\in B$) higher up by a "pipe". ...

**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 $...