# Tagged Questions

**-1**

votes

**2**answers

179 views

### Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...

**6**

votes

**1**answer

276 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

**0**answers

78 views

### An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...

**1**

vote

**1**answer

104 views

### Dependent Bernoulli sequence for which the strong law fails to hold

Background:
The strong law of large numbers (SLLN) is a powerful result in
probability, and there has been extensive literature on when the SLLN holds.
However, constructing nontrivial examples for ...

**1**

vote

**1**answer

73 views

### Upper tail concentration of sample covariance matrices

I'm interested in concentration of the following random matrix sum in spectral norm
$\frac{1}{m}\sum_{k=1}^m b_k^2\mathbf{a}_k\mathbf{a}_k^*$
Here $\mathbf{a}_k\in\mathbb{R}^n$ are i.i.d. standard ...

**3**

votes

**1**answer

179 views

### Expected visits to the origin by a symmetric random walk on the integers

Consider the first $2n$ steps of a simple random walk on the integers, starting at the origin. A simple binomial argument shows that regardless of $n$, the origin gets visited the most (in expectation)...

**3**

votes

**2**answers

118 views

### Extreme couplings

Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...

**1**

vote

**0**answers

65 views

### Concentration of the quotient of random variables

Let $X_1, X_2, \cdots, X_n$ be n i.i.d. standard Gaussian random variables. It is clear that we can describe the concentration of $\sum_{i=1}^n \alpha_i X_i$, and $\sum_{i=1}^n \alpha_i X_i^2$ (sub-...

**0**

votes

**0**answers

30 views

### Sufficient moment conditions to make $E[\sup_n |X_n|]< \infty$ for Markov process $X_n$

Is there any Markov process $X_n$ for which we can impose sufficient moment condition which will imply $E[\sup_n |X_n|]< \infty$

**3**

votes

**1**answer

174 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

121 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 \, ...

**0**

votes

**0**answers

34 views

### Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...

**3**

votes

**1**answer

135 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 \...

**1**

vote

**0**answers

46 views

### Expected number of forward jumps to reach a given quantile of a rv [closed]

I'm a noob in randomized algorithm and ran into a problem(definitely not home work. I'm doing a self study out of my interest with help of my friends. I'm pursuing research career in a machine ...

**3**

votes

**1**answer

106 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 ...

**1**

vote

**0**answers

46 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 $X$ ...

**0**

votes

**0**answers

90 views

### Bounds on Wasserstein (Kantorovich) distance

Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...

**2**

votes

**0**answers

708 views

### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required.
In random signal theory, this distribution is typically a ...

**3**

votes

**0**answers

86 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) ...

**0**

votes

**0**answers

67 views

### Refined versions of Azuma's inequality

Is there any version of Azuma's inequality where the bound $c_k$ as mentioned in https://en.wikipedia.org/wiki/Azuma's_inequality comes in the numerator of the fraction in the negative expoential.

**2**

votes

**1**answer

96 views

### An Inequality Regarding the Squared Conditional Variance

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. ...

**6**

votes

**0**answers

97 views

### Extreme unitary minimal models of conformal field theory

Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...

**2**

votes

**0**answers

71 views

### Factors between IID on trees: what about the useless information?

Let $p \in (0,1)$. Take $E$ to be the edge set of the trivalent tree $T$, and $G$ to be the automorphism group of $T$. Let $f$ be any $G$-equivariant map from the measure space $([0,1]^E, \text{d}x^{\...

**8**

votes

**1**answer

282 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

101 views

### Random Walk 2D with dependent weights [closed]

I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...

**1**

vote

**0**answers

75 views

### Increase mutual information for binary symmetric channel

I have a question about increasing mutual information for the binary channel. Assuming there is an independently $K$ dimensional binary source signal denoted by $X=[X_1, X_2, \cdots, X_K]$, a parallel ...

**0**

votes

**2**answers

112 views

### Different inner products for vector spaces of random variables

The inner product that appears in most books on probability is the covariance $\langle X,Y \rangle = E[XY]$ (considering that $X$ and $Y$ are zero mean real random variables).
Are there other inner ...

**1**

vote

**0**answers

113 views

### Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...

**0**

votes

**0**answers

69 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....

**1**

vote

**0**answers

136 views

### A weighted ergodic average

According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...

**3**

votes

**1**answer

96 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 ...

**0**

votes

**0**answers

57 views

### Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for
$$ E[\sup_{0 \leq n \leq k} X_{n}]$$
in terms of moments of $E[X_n], 0 \leq n \leq k$

**1**

vote

**1**answer

98 views

### Do there exist random variables that force transitivity of dependence? [closed]

In general, statistical dependence is not transitive. If $Y$ and $X_{1}$ are dependent, and $Y$ and $X_{2}$ are dependent, then $X_{1}$ and $X_{2}$ are NOT necessarily dependent.
However, in some ...

**3**

votes

**1**answer

88 views

### How to show monotonocity and the limit? [closed]

Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...

**2**

votes

**1**answer

148 views

### Probability of covering a set

Suppose we have a set of $N$ numbers. At any given trial we can randomly choose $N^{1-a}$ of the numbers where $a\in(0,1)$. We replace the numbers back.
How many trials does it take in average case ...

**3**

votes

**1**answer

171 views

### What's the best betting strategy to double money if we have $\delta$ advantage?

Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double ...

**0**

votes

**0**answers

23 views

### Characterisation of non-Gaussian stationary stochastic processes via auto-correlation functions

It is well-known that a centred stationary Gaussian stochastic process is characterised up to equivalence by its autocorrelation function.
Wiener, in his Time Series, makes the off-hand remark that ...

**0**

votes

**0**answers

26 views

### Unidirectional continuous path discrete time random walk

Is there any material available to study on unidirectional continuous path discrete time random walk on a line interval.
To say "unidirectional continuous path discrete time random walk on a line ...

**2**

votes

**0**answers

106 views

### Infinite total variation of complex measure in Feynman path integral [closed]

I am trying to understand this: If one tries to define a Feynman path integral as a Wiener integral, then the complex measure could be of infinite total variation. What exactly does this mean? How ...

**8**

votes

**11**answers

1k views

### What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....

**1**

vote

**0**answers

83 views

### Markov Chains and Simple Machine Learning [closed]

Suppose I have a large training set consisting of many strings of symbols.
$TS = \{Str_0, Str_1, ..., Str_n\}$
$Str_i = \{Sym_0 ... Sym_{len}\}$
These strings of symbols are each generated by the ...

**0**

votes

**0**answers

57 views

### CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...

**1**

vote

**2**answers

122 views

### Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s [closed]

I want to prove that in a sequence W of length n, consisting of 1s and 0s, $P$( in $W$ there is at most $\frac{\log_2n}2$ consecutive zeroes ) $\leq \frac{K}{n} $ for some constant K. Can anyone ...

**3**

votes

**1**answer

70 views

### Converging to moments obeying Carleman's condition

I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that $\mu_N$ are a sequence of measures on $\mathbb{R}$. Let $m_{N,k}$ be the $k$-th ...

**1**

vote

**1**answer

91 views

### If the sample space is an Euclidean Space, we can use a different type of PDF

Reading this post, I realize that is possible to have another type of PDF (probability density function) in the special case when the sample space is an Euclidean space.
Usually, we have a ...

**1**

vote

**0**answers

23 views

### Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map
$\theta \to I(\theta)$ where $I(\theta)$ is the ...

**1**

vote

**0**answers

61 views

### How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$
where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$
Initial condition:
\begin{eqnarray}
F(0, r) & = &...

**4**

votes

**2**answers

159 views

### Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...

**3**

votes

**1**answer

102 views

### Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
\sup_{t,s\in[0,1]}\frac{|W_t-W_s|}{|t-s|^{\...

**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 ...