All Questions
13 questions
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
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
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
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
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
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
$$
Y_i\sim\mathcal{...
- 103
2
votes
1
answer
521
views
Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
- 418
1
vote
1
answer
215
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
1
vote
0
answers
44
views
Small parameter expansion of probability density
I am trying to describe the motion of a particle that moves according to the Langevin equations
\begin{align}
\dot{x}&(t)=v_0\cos{\beta(t)},\tag{1}\\
\dot{y}&(t)=v_0\cos{\beta(t)},\tag{2}
\end{...
- 11
0
votes
1
answer
480
views
Brownian motion of every point in the plane
Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does ...
- 19
0
votes
1
answer
599
views
Random walk with exponential decay
A problem which arises in learning algorithms is $$x_{k+1}= \alpha x_k + \beta e_k$$
where $x_k$ is the scalar state variable at time $k$ and $e_k$ is an independent $\mathrm{Normal}(0,1)$ excitation ...
- 51