All Questions
9 questions with no upvoted or accepted answers
4
votes
0
answers
219
views
Conditional distribution of steps of random walk given the sum
Set-up. Consider a random walk $S_n=\sum_{i=1}^n X_i$, where $\{ X_i, 1\leq i < \infty \} $
is a sequence of i.i.d. random variables with distribution $\mu$, $\mathbb{E}X_1 = 0$. Let $a > 0$.
...
- 724
3
votes
0
answers
128
views
Probability that explosive random walk $X\to\gamma X+\epsilon$ with $\gamma>1$, always stays positive
Let $\{\epsilon_t\}_{t\ge0}$ be a sequence of iid random variables with full support. Let $\delta\ge0$ and $\gamma>1$. Then set $X_0 = \delta$ and define for $t\ge0$:
$$
X_{t+1} = \gamma X_{t} + \...
- 479
2
votes
0
answers
110
views
Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
- 4,432
2
votes
0
answers
153
views
Reference request for a result on subsets unlikely to be hit by random walks in a group
Suppose we are performing a random walk in a group. More precisely, we have a finite generating set $S$ of a group $G$ and the probability of walking along generator $s$ is given by $\mu(s)$ for some ...
- 21
1
vote
0
answers
91
views
A random process with conserved momentum: 'particle decay'?
Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...
- 5,038
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
1
vote
0
answers
262
views
Green's function for simple random walk after t steps in dimension 2
Let $p_{t,v}$ be the probability a discrete time, simple random walk at $v \in \mathbb Z^2$ reaches 0 within $t$ steps. We are looking for a reference to cite or an easy proof for the big-O behavior ...
- 457
1
vote
0
answers
60
views
Probability for a SRW to be at some place in an even number of steps
I am looking for some references for the following problem.
Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)_{t\...
- 166
0
votes
0
answers
115
views
Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?
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