All Questions
30 questions
4
votes
0
answers
112
views
MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem
Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define
$$X = a_1 X_1 + a_2 X_2 +...
- 637
4
votes
1
answer
518
views
Probability to return to the origin for a uniform random walk
Consider a uniform random walk on $\mathbb{R}$, with stepsize chosen uniformly from the interval $(-1,1)$. The random walk start at $x=0$. Denote by $\rho_p dx$ the probability that the random walk ...
- 188k
4
votes
1
answer
250
views
Does a subset with small cardinality represent the whole set?
Assume that we have heavy-tailed distribution $F(x)$ such that
\begin{align}
F(x)=\mathbb{P}[X\geq x]=x^{-0.5}.
\end{align}
Then, we produce $N$ independent samples $X_1,X_2,\ldots,X_N$ from this ...
- 287
4
votes
1
answer
338
views
Minimum of random walks
Let $M$ independent and identical random walks that follow the chi-squared distribution, i.e. in each step, a $X^2_1$ random variable is added. I am interested in the minimum random walk at each step. ...
1
vote
1
answer
215
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
4
votes
2
answers
480
views
Hitting probability of a line
Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ ...
- 17k
1
vote
0
answers
44
views
Small parameter expansion of probability density
I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align}
\dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\
\dot{y}&(t)=v_0\cos{\beta(t)},\tag{2}
\end{...
- 11
5
votes
0
answers
130
views
Random process on a sequence of rolls of an $n$-sided die
Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
- 83
10
votes
4
answers
680
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773
0
votes
0
answers
250
views
Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables
I asked this on MSE, but got no answer, hence asking here now. Help appreciated!
My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
- 1,467
2
votes
1
answer
900
views
Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$
Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following:
...
- 1,467
2
votes
1
answer
210
views
Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,467
1
vote
0
answers
83
views
Tracy Widom type results for asymptotic distribution of the $k$-th largest eigenvalue of the sample covariance when $n, p \to \infty$?
Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...
- 1,467
5
votes
1
answer
252
views
What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it?
Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|_1/\sigma)$$
be the pdf of the Laplace ...
- 53
3
votes
2
answers
229
views
Expectation of the exitpoint distance for the symmetric random walk
Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
- 31
2
votes
1
answer
228
views
Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v
Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and
$$M_n = \max_{1\leq k \leq n} S_k.$$
Is it ...
- 485
2
votes
0
answers
232
views
Random walk and comparing sums of Exponential random variables
Let $\sigma$ be the time a nearest neighbor random walk started at 1 that has probability $p>1/2$ of moving left reaches $0$. Let $\sigma'$ be an independent copy of $\sigma$. Let $(X_k)_1^\infty$ ...
- 457
7
votes
0
answers
774
views
Calculate the expectation of the maximum of averaged random walks
Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$
Is ...
- 773
4
votes
1
answer
288
views
Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups
Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (...
- 93
3
votes
1
answer
281
views
Stein's Equation for Gaussian Mixtures
In the paper "Spin glasses and Stein's method" (https://arxiv.org/pdf/0706.3500.pdf), Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes ...
- 1,127
3
votes
1
answer
196
views
Minimizer of two random walks
Consider the following two random walks:
The first random walk $\{S_n\}$ has i.i.d. step size
$$
X_i\sim\mathcal{N}(1,1)
$$
The second random walk $\{S'_n\}$ has i.i.d. step size
$$
Y_i\sim\mathcal{...
- 103
6
votes
0
answers
183
views
Distribution of the stopping time of an autoregressive sequence
Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...
- 1,127
5
votes
1
answer
297
views
Random walk with continuously distributed steps on [-1,1]
A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...
- 108
6
votes
2
answers
2k
views
Distribution of $\max_{n \ge 0} S_n$, random walk
Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...
- 61
1
vote
1
answer
378
views
Discrete random walk with uniformly distributed transition p, set initially
I've been working on a discrete version of the "unreliable friend" distribution. It would seem that what I've come up with is equivalent to the following random walk:
Choose $p$ from $U(0,1)$
Start ...
- 11
6
votes
1
answer
1k
views
Properties of a finite random walk
Consider the simplest random walk - $X_0 = 0$ and from there on (i.i.d), $X_i=X_{i-1}+1$ with probability $p$ or $X_{i-1}-1$ otherwise.
Let $Y_N$ be the highest point $X$ have reached on the first $N$...
- 618
8
votes
2
answers
3k
views
Expectation of Maximum of Uniform Multinomial Distribution
Suppose we have a uniform multinomial distribution with $k$ buckets, i.e. we put $n$ items uniformly at random in $k$ buckets leading to $n_1, \dots, n_k$ items in each bucket respectively. Let $m = \...
- 733
2
votes
1
answer
521
views
Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
- 418
5
votes
0
answers
1k
views
Compute the expected value of the next step of a sorted random walk
Here's what I'm thinking about. If you have a random walk (move +1 or -1 at each step) of some fixed length, then if you're at the maximum of the walk, the next step you take is -1 with probability 1. ...
5
votes
2
answers
3k
views
Probability of return at step $n$ of a Random walk to its starting vertex
Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the ...
- 65