For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

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79 views

1-d random walk probability bound calculation problem

I'm recently reading the paper about digital fountain code "LT Codes" by M. Luby. There is a statement seems simple with the author "The probability a random walk of length $k$ deviates form its mean ...
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1answer
210 views

Meaning of root mean square distance in random walk

This is a question about a simple random walk problem where we want to measure the average distance from the start in various experiments of N steps each. If d is distance moved during one such ...
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1answer
77 views

Strong Law of Large numbers, prove expression is Standard Normal

Question: "Let $X_{1},X_{2},\cdots$ be a sequence of independent random variables such that $X_{n}$ is binomial with parameters $2n-1$ and $p=\frac{1}{2}$. If $$Y_{n} = \frac{2(X_{1}+X_{2}+\cdots ...
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58 views

A matrix-multiplication random walk

Let $x \in \mathbb{R}^n$. Consider an $n\times n$ matrix $A$. Suppose we're interested in how $||A^nx||$ grows with $n$, the answer (excluding pathological cases) is that it scales exponentially with ...
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1answer
66 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
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1answer
344 views

Expected number of steps in a random walk with a boundary

Let's say I am trying to climb a flight of $N$ stairs. Each time I want to take a step, I flip a fair coin. Heads means I take a step up; tails means I take a step down. If I'm at the bottom of the ...
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26 views

Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
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56 views

Identities in proving the arcsine law

In the course of proving the Arcsine Law for 1-dimensional random walk, there appear two combinatorial identities: (We are always considering a simple symmetric random walk of length $2n$) 1. ...
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43 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
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1answer
71 views

First-passage probability with absorbing boundary at origin (No Laplace)

I have the following problem which I would like to solve without using Laplace transform. Can you possibly help or provide pointers? What is the first-passage probability, and mean first-passage time ...
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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 ...
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2answers
35 views

Computing the sum of a Catalyn sequence— Random-walk motivated

How would one go about computing the following?: $$\sum_{n=0}^\infty (.5)^{2n+1} \cdot \frac{{2n}\choose{n}}{n+1}$$ The motivation is that this gives the probability that a random walk on a number ...
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1answer
39 views

Proving that the eigenvectors of this class of matrices are the binomial coefficients

So I'm trying to figure out the behavior of this system: you have $N$ coins, and every step, you choose one of the coins randomly and flip them. Now we imagine a bazillion of these systems. We call ...
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0answers
61 views

Markov Chain Problem

I have been stuck on this question for days and really need some help. There are two methods, A and B, to finish a work. Method A succeeds with probability 1/3, but if it fails one tries method B ...
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1answer
32 views

Bounding the number of visits for each site of a random walk by a sequence

Recently, I asked if, for each $k>1$, a transient random walk visits each site less than $k$ times a.s.. You can find the question here: Visits from a transient random walker on the integers This ...
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1answer
64 views

Visits from a transient random walker on the integers

Consider a random walk $\{S_n\}$ on $\mathbb{Z}$ with forward probability $p>\frac12$. It is known for such a transient RW that each site is a.s. visited only finitely many times. However, is it ...
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2answers
69 views

Random walk on one-dimensional lattice - understanding the expression $pe^{i\theta} + qe^{-i\theta}$

I've started reading the book - First Steps in Random Walks and in the very first example in Chapter 1 they talk about a random walk on a one-dimensional lattice. If we consider a particle starting ...
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2answers
60 views

For a Gaussian Random walk where $x_n$ is the sum of $n$ normal random variables, what is $P(x_1 >0, x_2 >0)$?

I know that the events $x_1 >0$ and $x_2 >0$ are not independent, but I can't think of a way to find a conditional probability so I can solve this. Thanks!
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1answer
98 views

Recurrence for dependent random walks.

Let $\{X_i\}_{i\in\mathbb{N}}$ be a sequence of random variables taking values in $\{\pm e_1,\pm e_2\}$, where $\{e_1,e_2\}$ is the standard basis of $\mathbb{R}^2$. If $\{X_i\}$ are i.i.d. ...
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1answer
61 views

Martingales of random walk

Let $S_n$ be a random walk process defined by $$S_n=X_1+\dots+X_n$$ with $X_i \sim N(\mu,\sigma^2)$ and $X_i$ are i.i.d. I'm trying to prove that the quantity $(S_n-n\mu)^2-n\sigma^2$ is a ...
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110 views

How can we directly see that the number of random walks starting and ending at the origin is ${n\choose n/2}^2$?

In an infinite two-dimensional square-shaped grid, we define four directions, north, south, east, west. We thus have $4^n$ random walks of length $n$. If we end where we started, for every north step ...
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123 views

Recurrence of a certain class of $2$-$d$ random walks

As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for ...
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28 views

Sums of independent random variables

I was unsuccessful in deriving a good estimate of the distance below. Let $(X_{n})_{n \geqslant 1}$ be a sequence of i.i.d. random variables, and let $(\varepsilon_{n})_{n\geqslant 1}$ be a sequence ...
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46 views

Random Walk in confined region and loop configurations

Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go ...
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1answer
93 views

Solving a differential equation (Lattice Laplacian)

Suppose that $ p_n(t) $ is the probability of finding n particle at a time t. And the dynamics of the particle is described by this equation : $$ \frac{d}{dt} p_n(t) = \lambda \Delta p_n(t) $$ ...
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1answer
253 views

Random walks and diffusion limits

Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the ...
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3answers
89 views

A fly on a triangle?

A fly is on the vertex of a triangle. It can move left with probability $\frac 12$ and right with probability $\frac 12$. What is the expected number of moves till it reaches its starting point? ...
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2answers
198 views

Calculating expected value of random walk with one stop value.

I know that for a random walk with two stop values, the expected value of the number of steps needed is $ab$ where the stop values are $-a$ and $b$ and the initial position is at 0. What about for ...
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35 views

How to perform a stochastic search of the locality of a node in a network?

In a graph that may be a random graph (ER graph), scale free network, etc. I would like to obtain a distribution of the locality of the nodes surrounding a ...
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2answers
50 views

Random Walk with Edges

The setup for the specific problem that led to this question is as follows: You are playing a game at a casino and have \$10,000; The bank has \$2,000. You are making \$1,000 bets, with a equal ...
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1answer
322 views

Expected time for winning in biased Gambler's Ruin

Consider the random walk $X_0, X_1, X_2, \ldots$ on state space $S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with $P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where ...
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109 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
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3answers
141 views

Probability of having X more heads than tails for N tosses.

Giving a a fair coin, and tossing it N times, in how many possible outcomes would there be a point wherein there were more heads than tails tossed, ie, net heads.
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1answer
47 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 ...
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132 views

Brownian motion, rate of large events

Given the most simple brownian motion: $$ \dot x(t) = \sigma \eta(t)$$ where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$, I define as large event in a time-frame $\tau$ a portion of the trace ...
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101 views

Stopping time and martingale for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
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43 views

Random walk - proving limits exist

Consider the random walk $\{X_k\}_{k\geq0}$ on $\mathbb{Z}$ with transition probabilities $$\begin{cases} p_{i,i-1} = p_{-1} &> 0 \\ p_{i,i+1} = p_{1} &> 0 \\ p_{i,i+2} = p_{2} ...
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1answer
75 views

What is the probability that the robot steps on the bomb?

Suppose a robot is initially placed at $0$ on the number line, and is programmed to take steps of integer length in the positive direction between $1$ and $k$, inclusive, where $k$ is a positive ...
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3answers
215 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ with $p_1$, ...
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1answer
179 views

Probability of being at a certain point after $N$ steps in Random Walk with a single absorbing barrier

A random walker in $1$ dimension starts walking from a point $k>0$ with an absorbing barrier at point $0$. What is the probability that he will reach a point $m>0$ in $N$ steps? How should I ...
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0answers
92 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
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1answer
78 views

law of large number modified statement

The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$, $$ \forall ...
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1answer
997 views

Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ ...
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91 views

On the second derivative of Wiener process

As we all know, continuous white noise is the derivative, with respect to time, of a Wiener process. My question is that does the second derivative of Wiener process exists? If so, what is it and how ...
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1answer
97 views

Computation of a mean (random sum)

Let $X_1$, $X_2$, ... be independent and identially distributed positive random variables and define the sum $S_n = X_1 + X_2 + ... + X_n$. Consider the first time $N$ where $S_N \ge b$ with a given ...
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1answer
75 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 ...
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1answer
63 views

Absorbing state for a collection of random walks

Further to this question; having learned some stuff since I posed it. Consider a collection of random walks $X_i$ which take finite integer values. These evolve as time-inhomogeneous Markov Chains. ...
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2answers
57 views

Random walk confusion

If a ransom walk is binomial (1/2 probability of going forward, 1/2 backward) why isn;t the variance a) $\sigma=(\frac{n}{4})^.5$ b) instead of $\sigma=(n)^.5$ these sources seem to give ...
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28 views

What random process is this?

This is a rather basic question, is just that I don't know a thing about this subject. Let's say $x$ is an integer. At $t=0$, the value of $x$ is 1. Then, at each time step, one of the following ...
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1answer
76 views

Number non self avoiding closed walks surrounding some point

While studying some Peierls-like arguments in statistical physics I thought about the following problem: We have some 2d-integer lattice like this, for simplicity infinite in all directions. Now fix ...