All Questions
Tagged with random-walks stochastic-processes
124 questions
3
votes
1
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2k
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Hitting time probability in a Random Walk with possibility to die.
A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...
- 295
3
votes
1
answer
164
views
Simple linear asymptotics for leaving time of particle in open-boundary TASEP
EDIT: It appears the hypothesis may not be true - I am not sure. I therefore changed my question.
ORIGINAL QUESTION:
Consider a system $n$ linked discrete cells numbered $1 \ldots n$. Particles are ...
- 133
3
votes
1
answer
340
views
Importance resampling with exponential weighting
Suppose that we have
$$
\frac{p(x)}{q(x)} \propto \exp(\tau f(x)),
$$
where we can sample from $q$ but not from $p$. Our goal is to generate a set of particles $\{x_i\}_{i=1}^n$ such that $n^{-1}\sum_{...
- 1,127
3
votes
1
answer
884
views
Quadratic variation of sum of random variables
Let $N = (N_t)_{t\geq 0}$ be a Poisson process and consider random variables $Z_n$, $n\in N$. Compute the quadratic variations $[X]_t$ where $X_t = \sum_{n=1}^{N_t}Z_n$.
What I did was plugging $X_t$ ...
- 135
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
3
votes
1
answer
2k
views
first passage time, brownian motion
Hi,
If X(t) is Brownian motion in 2D, where X(0) = 0, then we can ask what is the expected time required to first hit a circle of radius R, centered at the origin. This is a First Passage Time ...
3
votes
0
answers
81
views
Random walk in a switching scenery
For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ ...
- 91
3
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0
answers
330
views
Random walk on $\mathbb{R}$ with "sticky" origin
Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...
- 313
2
votes
3
answers
300
views
How long does it take a Brownian particle to achieve a uniform probability distribution across a space?
Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" ...
- 599
2
votes
1
answer
280
views
Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
- 1,444
2
votes
1
answer
6k
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asymmetric random walk, hitting time probability
Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.
I am interested in the probability ...
- 295
2
votes
1
answer
847
views
Probability that a symmetric random walk returns to $0$ exactly $k$ times in $2n$ steps
I'm trying to find a formula to find the probability of exactly k returns in 2n steps of a symmetric random walk. More specifically, I am trying to show that the probability of 2 returns is exactly ...
- 21
2
votes
1
answer
192
views
Occupation time of non-stationary random walk
Assume $\varepsilon \in [0,1/2]$. Consider the discrete-time random walk $X_0 = 0$, $X_{t+1} - X_t \sim f(X_t) \delta_0 + (1-f(X_t))\operatorname{Rademacher}$, where $\delta_0$ is the Dirac delta on ...
2
votes
1
answer
460
views
Large deviation of random walk
1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk
$$
S_i = \sum_{j=1}^iX_j
$$
for $i=1,2,\ldots,n$.
I am looking for "good" exponential upper bounds ...
- 31
2
votes
1
answer
105
views
Convergence of a stochastic sequence?
I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...
- 185
2
votes
1
answer
528
views
Any modern/recent version of Ito & McKean?
This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
- 163
2
votes
1
answer
412
views
Does random walk have more concentration surrounding the origin?
Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval ...
- 502
2
votes
1
answer
521
views
Limit of a rescaled random sum of i.i.d. random variables
Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$
For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
- 418
2
votes
2
answers
167
views
Example of random walk in a random environment (RWRE) saying things on the environment
I was wondering if anyone is aware of works/articles/examples where random walks in a random environment (RWRE) are actually used for obtaining information on the random environment.
To clarify a bit, ...
- 59
2
votes
1
answer
184
views
A question about convergence of stochastic processes converging to a random walk
Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...
- 13
2
votes
1
answer
115
views
Randomly chosen walk of fixed length
Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$.
A walk of ...
- 138
2
votes
1
answer
168
views
Random Walk 2D with dependent weights [closed]
I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...
- 23
2
votes
1
answer
421
views
Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
- 340
2
votes
1
answer
447
views
MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
2
votes
0
answers
80
views
Bound from above and from below the probability that a 1-D centered random walk remains at each step inside a square root boundary
Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\...
- 63
2
votes
0
answers
114
views
Asymptotic Independence of random walks from increments?
Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
- 715
2
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0
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193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
2
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0
answers
78
views
A question on the convex hull of independent random walks
Consider $m$ independent random walks $X^1_n, \dots, X^m_n$ driven by a probability measure $\mu$ in $ \mathbb{Z}^d$. Assume that the $\mu$ has no drift, that is, the expected value of a $\mu$-...
- 6,224
2
votes
0
answers
63
views
hitting probabilities of oriented random walk
Consider a random walk on $\mathbb{Z}^2$, starting at $(0,0)$. Each step it moves rightwards with probability $p$ and upwards with probability $q=1-p$. The random walk terminates when it hits the ...
- 640
1
vote
1
answer
4k
views
First passage time of a 1D simple random walk in a discrete time infinite markov chain [closed]
If we consider a simple Random Walk on the positive integers (discrete Markov chain), with symmetric transition probabilities. We start at time $0$ at the integer $i_0 = m$ and at each time step $P(...
- 53
1
vote
1
answer
385
views
How fast does this Gaussian random walk move away from the origin?
Suppose $z_i$ are IID zero-centered $d$-dimensional Gaussian random variables with unit-trace covariance $\Sigma$ and $g(z_i)$ is the sum of its components.
Consider the following random walk:
$$x_s=\...
- 2,242
1
vote
2
answers
169
views
Asymptotic properties of weighted random walks / infinite convolutions of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum_{k=1}^n c^k X_k.
$$
I can prove that this ...
1
vote
1
answer
425
views
Invariance principle: Brownian bridge and random walk conditioned on end point
Let $\{X_i, i \in \mathbb{N}\}$ be a sequence of non-lattice i.i.d. centered random variables, $\mathbb{E} |X_1| ^3 < 0$. Let $S_n = \sum\limits _{i=1} ^n X_i$ be the corresponding random walk and ...
- 724
1
vote
2
answers
356
views
Asymptotic behavior of a random geometric sum
Let $S_n$ denote a simple random walk with i.i.d. increments $X_i$ such that $P(X_1 = 0) = P(X_1=1) = 1/2$, i.e. $$S_0 = 0, \ S_n = X_1 + \dots + X_n.$$
The behavior of $S_n$ as $n \to \infty$ is ...
- 121
1
vote
1
answer
247
views
Arc Sine law for Random Walk conditioned to non-absorption or not?
Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$.
Is the $E(S^{2}_{n}| \tau \geq n)$ known?...
- 61
1
vote
2
answers
772
views
Gibbs sampling step size
I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that ...
- 87
1
vote
1
answer
120
views
multi-time limit of a maximum of random walks
Suppose one has $N$ iid random walks $X^{(1)}_t,\ldots,X^{(N)}_t$ in discrete or continuous time $t$, let us say for example Poisson jump processes, and consider the stochastic process $Y^{(N)}_t = \...
- 568
1
vote
1
answer
90
views
A scaled random walk on the number line
An agent $A$ is performing a random walk on the number line. Let $X_t$ be his position at time $t$. $X_{t+1}$ is calculated according to the following rules:-
$ X_{t+1} =$
\begin{cases}
...
1
vote
2
answers
302
views
how to derive stationary distribution of maximal entropy random walk
I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...
- 211
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
2
answers
88
views
Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$
Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (-1-t_0,1-t_0)$, ...
- 6,853
1
vote
1
answer
233
views
Random walks on Galton–Watson trees
I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\...
- 39
1
vote
1
answer
276
views
Number of deaths in birth-death process conditioned on start and end points
Say I have a simple linear continuous time birth-death process with state space the non-negative integers, where there are parameters $b$ and $d$, with the rate (as you'd see in a $Q$ matrix) of going ...
- 11
1
vote
0
answers
41
views
Asymptotic mixing time and Euclidean probability distance for path graphs
We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\...
- 1,677
1
vote
1
answer
284
views
Rate of convergence to uniform distribution
Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid ...
- 6,853
1
vote
1
answer
109
views
Phase space Brownian bridge
I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$
s.t. $x_0=0$ and $x_1=0$
where $dw_t$ is a Wiener process.
I am thinking about ...
- 13
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
181
views
Random walk on 2d lattice with obstacles
Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...
- 6,853