## New answers tagged pr.probability

5
votes

### Limit distribution of the self-normalized sum of Cauchy random variables

It converges to 0 in probability. $\frac{\Sigma_1^nX_i}n $ always has the same cauchy distribution while $\frac{\Sigma_1^n|X_i|}n \rightarrow \infty $ by the law of large numbers

2
votes

### Upper-bound of the tail of a weighted sum of iid random variables

This is a partial answer, in the regime that the probability in question is close to $1$. I normalize so that $EY_i=1$. The example $a_i=1/\sqrt{n}$ shows that you need to take $t\geq \sqrt{n}/2+O(1)$...

3
votes

### The probability that iid draws from a mean zero random variable sum to zero

I guess I can convert my comment to an answer.
Recall that for a bivariate power series $F(x,y) = \sum_{i,j \geq 0} f(i,j) x^i y^j$, its diagonal is the univariate power series $\operatorname{diag} F(...

Community wiki

3
votes

### Optimally betting a beta-biased coin

Next and let's hope final attempt, after numerous mistakes, a lot of nonsense, and an obscene number of edits: I think I have circumstantial evidence that things will just be very messy and the ...

0
votes

### On stochastic integration

Jacod/Shiryaev mention in the next paragraph "a fundamental result by Bichteler, Dellacherie and Mokobodzki, which explains why the space of semimartingales is so important.". The Bichteler-...

2
votes

Accepted

### Reference request: Gaussian measures on duals of nuclear spaces

There is a choice to be made here: working with Banach spaces or with spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$. There are pros and cons for each of these two settings.
Let me stick to ...

1
vote

Accepted

### Stochastic order on weighted sum of iid random variables

This conjecture does not hold, even for $n=2$.
Indeed, then the difference between the left-hand side of your inequality and its right-hand side is
$$d(a):=g(a)-g(1/\sqrt2),$$
where
\begin{equation*}
...

5
votes

Accepted

### Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?

This is basically the argument in Sean Eberhard’s comment posted while I was writing this answer.
Note that the condition on $A$ is exactly the same as: $A$ follows a uniform distribution on $\mathcal{...

0
votes

### Expectation of top-K selection of squared Gaussian random variables

Let $Z \sim \mathcal{N}(0,\sigma^2 I_n)$ be an $n$-dimensional Gaussian vector with independent $\mathcal{N}(0,\sigma^2)$ components. Consider the family of binary vectors in $\mathbb{R}^n$ with ...

1
vote

### In an Erdős–Rényi random graph, what is the threshold for the property "every edge is contained in at least one triangle"?

Let $G$ be a random graph on $n$ vertices where each edge exists independently with probability $p = \frac{c}{n}$ for $c > 1$. The total number of edges in a complete graph on $n$ vertices is $m = ...

10
votes

Accepted

### How to optimally bet on a biased coin?

The strategy suggested in Geoffry Irving's answer of betting the maximum amount each time is correct, but the argument given is incomplete. The expected final amount, conditional on the outcomes of ...

1
vote

Accepted

### A Kolmogorov inequality for sums of contiguous subsequences

If $p_i:=P(X_i=0)>0$ for all $i$, then
$$P_n(\lambda):=P(\max_{1\le j < k \le n}|S_k - S_j| < \lambda)\ge P(X_1=0,\dots,X_n=0)=p_1\cdots p_n>0$$
for all $\lambda>0$, so that $P_n(\...

7
votes

### How to optimally bet on a biased coin?

It is optimal to always bet the full amount iff heads has come up most of the time, and zero otherwise (breaking ties arbitrarily).
To see this, first note that any strategy can be scaled linearly, so ...

5
votes

### Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?

Such a transition from the discrete to the continuous is precisely the point of Nelson's Radically Elementary Probability Theory (REPT). It recently turned out that when viewed as a subsystem of BST,...

0
votes

### Deriving the distribution of standardized variables with empirical mean and standard deviation

Anthony's comment has the correct answer: $Z=(z_1,\ldots ,z_N)$ is uniformly distributed on $\|Z\|^2=N$, $\langle e, Z\rangle=0$, $e=(1,1,\ldots, 1)$ (that is, the joint distribution is the $(N-2)$-...

1
vote

Accepted

### Weak convergence of random measures generated by non-negative martingales?

Partial answer
For every $a \in [0,1]$, $(\mu_n([0,a]))_{n \ge 0}$ is still a non-negative martingale, hence it converges almost surely to some random variable $L_a$ with values in $[0,+\infty]$. One ...

3
votes

Accepted

### Show convergence result

$\newcommand\de\delta\newcommand\ep\varepsilon$You wrote
Could you help me to show that under Ass1 and Ass2
$$d_H(A, A_n)\rightarrow_{a.s.} 0$$
Of course, this is not true in such generality. For ...

0
votes

### Show convergence result

EDIT: That whole answer assumes the $p_n$'s are uniformly and independently distributed on $[0,1]$. I have no idea on how to extend it to the general case of other distributions.
It gets elementary ...

1
vote

### Minimum of exponential distribution

$\newcommand\la\lambda$Conditioning on $X_1$ and using the independence of the $X_j$'s, we have
$$\begin{aligned}
P(X_1<X_2,\dots,X_n)
&=\int_0^\infty P(X_1\in dx)\,P(x<X_2,\dots,X_n) \\
&...

0
votes

Accepted

### Existence of the limit of periodic measures

I may be misunderstanding the question, but in full generality, the answers are "no," i.e. the $\mu_p$ can fail to converge, and even when they converge to a limit $\mu$, $f(\mu_p)$ can fail ...

1
vote

Accepted

### Probability problem in Sheehan's conjecture

From the paper Independent Dominating Sets and Hamiltonian Cycles by Haxell, Seamone, and Verstraete, if you look at Hamiltonian cycle of length $4n$, along with $n$ random disjoint copies of $C_4$, ...

0
votes

### Expectation of the trace of inverse of a Gaussian random matrix

The following argument is quite similar to Carlo Beenakker's.
For simplicity, I only consider the real case. I'm also going to use different symbols for the sizes of the matrices. Let $X$ be an $n \...

0
votes

Accepted

### Projection of an element of the $n$-simplex onto subset

$\newcommand\S{\mathbb S}$Let $|\cdot|$ denote the Euclidean distance and let $k\ge1$ denote the cardinality of $K$. Then for any $x\in\S^n$ and $y\in\S_n(K)$
$$|y-x|^2=\sum_{j\notin K}x_j^2+\sum_{j\...

3
votes

Accepted

### What happens when the diffusion term in an SDE becomes zero?

Following a comparison result (eg. Comparison theorem in Revuz-Yor Chap. IX §3), we will show that $X_{t}\leq b$ for all $t\geq 0$. This result requires though the Yamada-Watanabe regularity
$$|\sigma(...

3
votes

### How does pairwise independence restrict dependence to a third variable?

Let $X,Y$ by i.i.d. with $\Pr(X=+1) = \Pr(X=-1) = 1/2,$ and let $Z=XY.$ Then $Z$ has the same distribution as $X$ or $Y,$ and $X$ and $Y$ are independent and $X$ and $Z$ are independent and $Y$ and $Z$...

4
votes

Accepted

### How does pairwise independence restrict dependence to a third variable?

No, this is is hopeless, as becomes clear when you write
$$
\operatorname{Cov}(X,Z)+\operatorname{Cov}(Y,Z)=E(X+Y)Z . \tag{1}\label{463874_1}
$$
Your assumptions don't restrict $U=X+Y$ much, you only ...

7
votes

Accepted

### Hopf monads in categorical probability theory

Since the idea of a probability monad is not a fully formal concept, there can be no fully formal answer to this question. Nevertheless, I will argue below that no nontrivial probability monad is a ...

2
votes

### Expected number of coin flips before you see a $k$-term arithmetic progression of heads

See Maximal Arithmetic Progressions in Random Subsets by Benjamini, Yadin and Zeitouni, ECP 12: 365-376 (2007) and the erratum in ECP 17: 1-1 (2012). See also the extensions in
M.-Z. Zhao and H.-Z. ...

1
vote

Accepted

### Lower bounds for truncated moments of Gaussian measures on Hilbert space

Here we will provide a necessary and sufficient condition for $I_k$ and $J_k$ to be bounded away from $0$.
We have
\begin{equation}
I_k=E\|a_k^{1/2}X\|^n\,1(a_k^{1/2}\|X\|<r_k),
\end{equation}
...

1
vote

### Non-adjacent permutations

Too big to comment, so I'm writing this here.
If we consider the $1$-dimensional circle ($S^1$) of length $n$, then the permutations $\pi$ which disallows $i \rightarrow i+1$ ($i \rightarrow i-1$ is ...

2
votes

### Simple linear asymptotics for leaving time of particle in open-boundary TASEP

The $O(n)$ statement in your original question is true. (For the $O(j)$ statement, see the note at the bottom.)
If $n$ is your parameter defining the size of the system, and $k\ge1$ is another ...

3
votes

Accepted

### Evolution of the empirical mean of a list as we remove elements proportional to their value

A simple mean field calculation suggests to look at the solution to
$$
\partial_t Q_t(k) = -\frac{k Q_t(k)}{\sum_\ell \ell Q_t(\ell)},\qquad Q_0 = P.
$$
For all $n$, one would then expect a good ...

1
vote

Accepted

### Concentration for sum of order statistics

$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\D}{\overset{D}=}$Let $X_{n:1},\dots,X_{n:n}$ denote the order statistics in question. Fix any real $t>0$. You wanted to show that ...

0
votes

### Winning game probability

Simply discard when we get draws, as there are identical sub-problems with the original one.
Compute straightforward the probability when Alice wins by how many alternating $a,b$ groups we have:
$(1+b+...

1
vote

Accepted

### Property of $p$-norm in the $n$-simplex

The answer is yes if $p=2$ (by the Pythagoras theorem: $\|x\|_2^2=\|x-u\|_2^2+\|u\|_2^2$, because $x-u\perp u$) and, obviously, if $p=1$ or $n\le2$.
Otherwise, the answer is no, because then the value ...

6
votes

### Winning game probability

Here goes a combinatorial proof of the answer (formula (1)) by Christian Remling.
If Alice wins, then after the last draw (or after the beginning if there were no draws) we have a sequence $B^jTA^N$, ...

6
votes

### Winning game probability

As indicated in the comments, this is not difficult in principle, but the details seem tedious. Alice wins with probability
$$
\frac{a^NS_N(b)}{a^NS_N(b)+b^NS_N(a)} , \tag1
$$
with
$$
S_N(x)=1+x+\...

1
vote

### Functional dependence not preserved in the weak limit

Let $X=[0,1]$ the unit interval.
For every $n\in\mathbb N$ let $a_{n,1},\dots a_{n,n}$ distinct points in $[0,\frac1n]$, chosen such that all $a_{n,k}$ are pairwise different.
Outside the set $A=\{a_{...

2
votes

### Functional dependence not preserved in the weak limit

Let $M=[0,1]$, $f$ is the indicator function of $[0,1/2)$, and $\mu_n=0.5 \delta_{1/2-1/2n}+0.5 \delta_{1/2+1/2n}$. Then $\mu=\delta_{1/2}$ but $\nu$ is uniform on $\{0,1\}$.

1
vote

### Evolution of the empirical mean of a list as we remove elements proportional to their value

I doubt this is an easy analysis, as the result is going to be affected by the distribution used for $P(k)$. I doubt there is a general form, but empirically there may be a well-behaved example.
In ...

3
votes

### Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$

Let $\mathrm{tr}:GF(p^n)\rightarrow GF(p)$ be the trace function which is equidistributed. Choose a basis so that you consider $GF(p^n)$ as the vector space $GF(p^n)$. For any $c \in GF(p^n)^\ast,$ ...

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