-1
votes
0answers
30 views

Random walk probability [on hold]

A particle executes a simple unrestricted random walk on a straight path, a step to the right of length 1 occurring with probability 1/3 and a step to the left of length 1 occurring with probability ...
1
vote
0answers
22 views

Conditional return time of simple random walk

Consider a simple random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0 = 0$. The probability to jump to the right neighbour is $p \geq \frac{1}{2}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, ...
7
votes
1answer
42 views

Random walk on cubic lattice

Suppose at every point of the cubic grid in n dimensions is a particle, and at every timestep every particle moves at random to one of its 2n neighbours. As time goes to infinity, what is the ...
0
votes
0answers
26 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
0
votes
1answer
26 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
0
votes
0answers
23 views

Weighted random walk in 1-dimension

Suppose we have random walker on a line, he can only stay on sites which are, say, a distance $a$ from each other. At each step he can go left or right. Every time he steps on a site, makes the ...
2
votes
1answer
58 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
0
votes
0answers
43 views

Proof of “changes of sign” in one-dimensional random walk model [Feller's section 3.5, page 84]

Consider the one-dimensional random walk of a particle. We shall denote the individual steps by $X_1, X_2, \cdots$ with $X_i = \pm 1$ and the positions by $S_1, S_2, \cdots$ with $S_i = X_1 + X_2 + ...
3
votes
1answer
84 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
3
votes
1answer
67 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
0
votes
1answer
62 views

2 dimensional random walk - hit of targets

Consider a random walk in $\mathbb{Z}^2$, $x(j) = x(j-1) + \xi_j$, where the increments are random variables independent and identically distributed with finite support, the expectation $m := ...
2
votes
1answer
76 views

central limit theorem for high dimensional random walk

Consider random walk in $\mathbb{Z}^d$, $d>1$, with $x(t) = x(t-1) + \xi$, where $\xi$ has some probability distribution in $\mathbb{Z}^d$ with finite support, expectation $m = \sum_{v \in ...
1
vote
1answer
49 views

ruin of the gambler with probability to die

Consider a random walk on $\mathbb{Z}$ starting from $i >0$. With probability $p$ it moves to the nearest neighbor on the left, with the same probability it moves to the nearest neighbor on the ...
2
votes
0answers
136 views

Finding functions where the increase over a random interval is Poisson distributed

I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function ...
1
vote
1answer
48 views

Concerning the distribution of a random variable of a random walk that doesn't make any sense to me

Let $\Omega = \{w = (x_1, \dots, x_N) | \; x_i \in \{-1, 1\}\}, \;X_k(w) = x_k, \;S_n(w) = \sum_{k=1}^n X_k(w), \; S_0(w) = 0.$ After having proven a few theorems about $S_n$, in our lecture about ...
2
votes
1answer
76 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
1
vote
1answer
58 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
3
votes
1answer
130 views

successive doubling the stake until head appears

I consider the following gaming system: Start with 1 dollar and always bet on head (coin tossing). You always double your stake until the first head appears. Maximum rounds: $n$ I formulated it as a ...
1
vote
1answer
35 views

Off-lattice Brownian bridges in R^3

Start at a point $(0,0,z_0)$ and take $n$ steps of unit length in a random direction (for each step) in $\mathbb{R}^3$. Let such a walk be valid if the position of the last step, and only the last ...
2
votes
3answers
106 views

Derivation of Wiener process first passage times using probability generating function?

I would like to find the distribution of first passage times in a simple Wiener process using the idea of probability generating functions. Thus there will be, at certain point, a limiting step to go ...
2
votes
0answers
207 views

Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
0
votes
0answers
119 views

Nonexistence of invariant probability with respect to random walk

We have a random walk process $X_{k+1} = X_{k} + Y_{k}$, where $Y_k$ are independent random values with the same distribution, $X_{0}$ have some fixed distribution $\pi$. Let $P^{k,s}(x,B)$ be a ...