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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
mike's user avatar
  • 1,129
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 $t\geq \sqrt{n}/2+O(1)$...
ofer zeitouni's user avatar
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(...
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 ...
Christian Remling's user avatar
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-...
Thomas Kojar's user avatar
  • 4,359
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 ...
Abdelmalek Abdesselam's user avatar
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*} ...
Iosif Pinelis's user avatar
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{...
Aphelli's user avatar
  • 288
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 ...
Damien's user avatar
  • 156
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 = ...
Damien's user avatar
  • 156
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 ...
Will Sawin's user avatar
  • 133k
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(\...
Iosif Pinelis's user avatar
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 ...
Geoffrey Irving's user avatar
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,...
Mikhail Katz's user avatar
  • 15.2k
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 $(N-2)$-...
Christian Remling's user avatar
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 ...
Christophe Leuridan's user avatar
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 ...
Iosif Pinelis's user avatar
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 ...
Claude Chaunier's user avatar
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) \\ &...
Iosif Pinelis's user avatar
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 ...
Ronnie Pavlov's user avatar
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 $C_4$, ...
Zach Hunter's user avatar
  • 3,335
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 \...
dohmatob's user avatar
  • 6,686
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\...
Iosif Pinelis's user avatar
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(...
Thomas Kojar's user avatar
  • 4,359
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 $Z$...
Michael Hardy's user avatar
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 ...
Christian Remling's user avatar
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 ...
Tobias Fritz's user avatar
  • 5,775
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. ...
ofer zeitouni's user avatar
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} ...
Iosif Pinelis's user avatar
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 ...
Alapan Das's user avatar
  • 1,755
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 ...
Dan Romik's user avatar
  • 2,470
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 ...
Martin Hairer's user avatar
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 ...
Iosif Pinelis's user avatar
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+...
heartwork's user avatar
  • 385
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 ...
Iosif Pinelis's user avatar
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 $B^jTA^N$, ...
Fedor Petrov's user avatar
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+\...
Christian Remling's user avatar
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_{...
Nandor's user avatar
  • 247
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 $\{0,1\}$.
Michael Greinecker's user avatar
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 ...
Henry's user avatar
  • 830
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 $c \in GF(p^n)^\ast,$ ...
kodlu's user avatar
  • 10k

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