All Questions
Tagged with random-walks stochastic-processes
124 questions
1
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44
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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
1
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0
answers
289
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Growth rate of exponential sum of $S_j$
Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$.
I'...
- 85
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76
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Spitzer's condition, a slowly varying function and its behavior
Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...
- 121
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96
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Minima of a random walk and an equality for a fraction
Let $S_n := X_1 + \dots + X_n$ denote a random walk with zero mean and finite variance and write $L_n := \min \{ 0, S_1, \dots, S_n\}$. The tail distribution of $L_n$ are well-known and in particular,
...
- 123
1
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1
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456
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Random walk with gaussian increments - Probability that it falls below 0
Suppose $\{Z_{i}\}_{i=1,2,\ldots}$ are normally distributed (identically and independent) random variables with mean $\mu>0$ and positive variance $\sigma^{2}$. Suppose we want to calculate the ...
- 13
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60
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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
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365
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Diagonal of Green's Function
I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....
- 185
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0
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309
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Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$
Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...
user71202
1
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1
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971
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Integration of independent Brownian motions
I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where $\tilde{...
- 11
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1
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480
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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
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1
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336
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How far does a random walker travel before returning to the origin?
Consider a (continuous time) simple symmetric random walk on $\mathbb Z$, starting from the origin. Let us denote this random walk by $\{X_t: t\geq 0\} $. It is well known that this random walk is ...
- 59
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2
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3k
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Two dimensional brownian motion first passage time
Hello,
I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion.
any papers, references, books or web links for study will be helpful.
thanks
...
- 21
0
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2
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266
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Last crossing of a line by a random walk
Let $X_1, X_2, ...$ be i.i.d. random variables, $\mathbb{E} X_1 > 0$, and let $S_n = \sum\limits _{i = 1} ^n X_i$. Define $\tau = \max \{n \in \mathbb{N}: S_n \leq 0 \}$ with the convention $\tau =...
- 724
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1
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77
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Reference request: $\mathbb{E}|X_t| \to \infty$ as $t \to \infty$ when $\{X_t\}_{t\geq 0}$ is a continuous-time (symmetric) random walk
Let $\{X_t\}_{t\geq 0}$ be a one dimensional continuous-time (symmetric) random walk on $\mathbb Z$ defined via $$X_t = X_0 + \sum_{i=1}^{N_t} Y_i,$$ where $X_0 \in \mathbb{Z}_+$ is a non-negative ...
- 730
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1
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110
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Skip-free random walks: recurrence and transience
Let us define a one dimensional random walk: for all $n\in\mathbb{N}$
$$
X_n:=\sum_{i=1}^nZ_i
$$
with $Z_i$ i.i.d. random variables taking values in $\{-1,0,1,2,\dots\}$. This process is sometimes ...
- 59
0
votes
1
answer
599
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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
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1
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160
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Probability to cross an envelopp for 1D random walk?
Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...
- 5
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1
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183
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Probability to cross dynamic boundary for 1D-random walk?
context: Imagine we have an evolving bit sequence (ex: 001011...) where the probability to get 0 or 1 is 1/2. n is the lengh of my sequence (the number of bits)
I can make an analogy with random walk: ...
- 5
0
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1
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613
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2 Random Walkers on 2d square lattice, Torus
I am looking for the probability that two random walkers initially at different sites, meet at step t if they are moving on a 2-dimensional torus(Square Lattice)
Any help would be appreciated.
- 53
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1
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239
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Transition probabilities for the symmetric random walk on the integers
I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
- 243
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0
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85
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Does a 2d random walk hit 0 for increasing distances AND time spans?
Question: For a simple symmetric random walk $(Z_t)_{t\geq 0}$ in $\mathbb{Z}^2$, does
$$\lim_{\beta\rightarrow 0}\mathbb{P}^{x_\beta}(Z_t=0\text{ for some }t\leq h(\beta)T)=0\quad (2.8)$$
where $|x_\...
- 31
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1
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635
views
Mean square displacement for a random walker in a finite system
It is well known that for a simple random walk on a 2D square lattice extending to infinity the mean square displacement of the walk $\langle \mathbf r^2\rangle \propto N \, :(*)$ with $N$ the number ...
user88381
0
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1
answer
150
views
Weak convergence of process
Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) \...
- 243
0
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0
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111
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Markov chains on a polyhedron
A modification of a question from Gerard Letac (1976): A m-sided q-adjacent-faced polyhedron has one of its faces "up." Each round, the polyhedron rolls so that any of the adjacent faces is now up. ...
