**1**

vote

**0**answers

99 views

### Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...

**1**

vote

**1**answer

58 views

### “Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...

**8**

votes

**2**answers

198 views

### Constructing an independent uniform random variable from two independent ones

Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable ...

**2**

votes

**0**answers

98 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 ...

**17**

votes

**3**answers

711 views

### Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality:
$$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$
where $X>e$ has ...

**2**

votes

**1**answer

74 views

### Do product distributions (or graph products) eventually cluster as more products are taken?

Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...

**0**

votes

**0**answers

108 views

### Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...

**4**

votes

**1**answer

168 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

158 views

### Calculate the KL divergence between two transition matrices

I want to calculate how different two markov transition matrices are.
For example:
$\begin{pmatrix} .2 & .8 \\ .1 & .9 \end{pmatrix}$
and
$\begin{pmatrix} .3 & .7 \\ .1 & .9 ...

**0**

votes

**2**answers

53 views

### A way to possibly calculate one Binomial CDF function from another closely related one?

Let $y < z$ be two numbers between $0$ and $1$, is there a way to relate the CDF functions $F_{n,y}(s)$ and $F_{n,z}(s)$... or approximate one from another, without just saying $F_{n,z}(s) \le ...

**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 ...

**4**

votes

**1**answer

188 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 ...

**6**

votes

**1**answer

288 views

### Hypothesis test beyond simple hypotheses (mathematical statistics)

In mathematical statistics, the following problem (simple hypothesis test) is considered: given a data sample, test the hypothesis $H_0$ stating that all sampled values are values of a random variable ...

**1**

vote

**0**answers

61 views

### Relation between Aitchison Distance on a Simplex and Geodesic distance on the multinomial manifold [closed]

I am trying to understand the difference/relation between the Aitchison distance on a simplex
$$\left[ \sum^D_{k=1} (\log{\frac{x_{ik}}{g(\mathbf{x}_i)}} - \log{\frac{x_{jk}}{g(\mathbf{x}_j)}})^2 ...

**0**

votes

**0**answers

78 views

### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...

**1**

vote

**0**answers

123 views

### Chain Rule for Maximal Correlation

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**1**

vote

**1**answer

86 views

### An inequality for Maximal Correlation over a Markov Chain

Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...

**1**

vote

**0**answers

61 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 ...

**1**

vote

**1**answer

111 views

### Independence of two random variable

Let $W$ and $S$ are two positive valued continuous random variable. Suppose
$g: [0,\infty)\rightarrow [0,\infty)$ is a convex function with a constraint that $g$ can't be of the form $g(x)=cx$, $c$ ...

**10**

votes

**1**answer

168 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 ...

**3**

votes

**0**answers

39 views

### Bound on principal angle of uniform random subspaces of different dimensions?

This paper derives the distribution of the largest principal angle between two subspaces sampled (independently) uniformly from the Grassmanian manifold of $p$-dimensional subspaces in $\mathbb{R}^d$, ...

**5**

votes

**1**answer

311 views

### An Inequality of KL Divergence

Given two probability distributions $P$ and $Q$ defined over a finite set $\mathcal{X}$, one can define the KL divergence between $P$ and $Q$ as
$$D(P||Q):=\sum_{x\in ...

**0**

votes

**0**answers

80 views

### How to prove a CAN estimator is also root n-consistent?

Given a consistent asymptotically normal estimator $\hat{\theta_n}$ of an unknown parameter $\theta$, how to prove that $\hat{\theta_n}$ is also a $\sqrt{n}$-consistent estimator of $\theta$?
...

**2**

votes

**1**answer

91 views

### Convergence of a test statistic

I'm reading a paper of Shao and Zhang:
Testing for Change Points in Time series.
In this paper they claim the following:
The are testing whether there is a change in the mean of a time series. So
...

**4**

votes

**2**answers

198 views

### Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$:
$$
\gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0.
$$
It is guaranteed to be positive definite by Polya's criterion ...

**1**

vote

**1**answer

130 views

### Large deviations for sums of random variables whose correlation function decays exponentially

Let $X_1, \ldots, X_N$ be a string random variables taking values $X_i \in [-1,1]$ and jointly distributed according $P(X_1, X_2, \ldots, X_{N-1}, X_N)$, which is invariant under cyclic permutations ...

**1**

vote

**0**answers

59 views

### Characterization of certain families of functions

For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon ...

**10**

votes

**1**answer

259 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 $\Phi$ ...

**5**

votes

**2**answers

129 views

### Gaps between descending order statistics

Let $\{X_{1},X_{2},\cdots,X_{n}\}$ be a random sample of size $n$. Denote $(X_{(1)},X_{(2)},\cdots,X_{(n)})$ to be its descending order statistics. Define gap $g_{i}(n)$ to be ...

**1**

vote

**0**answers

97 views

### Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...

**3**

votes

**1**answer

111 views

### $\int_0^t f(s)\,dB_s$ normally distributed, mean and variance

Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...

**1**

vote

**1**answer

92 views

### KL divergence Inequality

I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading.
Consider two probability measures $P$ and $Q$ both ...

**1**

vote

**0**answers

37 views

### Adding weights to the Brier score

Fix $n > 0$, and consider the space $\cal P$ of probability functions defined over the Boolean closure of a fixed $\cal S = \{ s_1, \ldots, s_n \}$. The Brier score of $P \in \cal P$ at $s_i \in ...

**2**

votes

**0**answers

124 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**3**

votes

**1**answer

137 views

### Practical bounds for the Wasserstein distance in 2 dimensions

Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let ...

**5**

votes

**2**answers

149 views

### Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let ...

**0**

votes

**0**answers

32 views

### Consistency of M-estimators when the constraint set also has to be estimated

Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
...

**2**

votes

**1**answer

111 views

### Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...

**0**

votes

**0**answers

38 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**0**

votes

**0**answers

32 views

### A book on discriminant analysis

Can anyone suggest a good book on discriminant analysis - comprehensible and detailed? (Kendall and Stuart write about the subject too concisely.)
Thanks in advance.

**0**

votes

**0**answers

181 views

### Best measure for curve similarity

I would like to measure similarity between two curves represented by two arrays of points.
The similarity measure should not depend on the size of these shapes. Two similar shapes but have different ...

**1**

vote

**1**answer

59 views

### Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...

**0**

votes

**0**answers

102 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**1**

vote

**2**answers

127 views

### Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...

**-2**

votes

**1**answer

83 views

### About the boundary conditions of the Black-Scholes-Merton PDE [closed]

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve.
Let $c(t,x)$ be the value of the ...

**0**

votes

**0**answers

46 views

### convergence of empirical distribution of random vectors

Given
(a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal
entries $A_{ij}\sim \mathcal N(0, 1/n)$; and
(b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...

**5**

votes

**0**answers

167 views

### Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...

**2**

votes

**1**answer

123 views

### Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...

**0**

votes

**0**answers

41 views

### Mixture model: optimization vs regression

Consider a sample $\mathcal D = \{T_n\}_{n=1}^N$ of independent random variables, s.t.:
$$
p(T_n) = p_n(T) = \sum _{m=1}^Mp_n(\mathcal C_m)p_n(T\mid \mathcal C_m) = \sum _{m=1}^Mw_{nm}q_m(T)
$$
I will ...

**0**

votes

**0**answers

26 views

### K nearest neighbors estimation with a kernel

If I have a bunch of data points $x_1,\dots,x_n$, I can build a density function $f(x)$ based on these data points by defining $f(x) = c/d_k(x)$ for an appropriate constant $c$, where $d_k(x)$ is the ...