Questions tagged [random-walk]

A stochastic process that describes a path arising from a succession of random steps.

Filter by
Sorted by
Tagged with
0
votes
1answer
1k views

Can we forecast a random-walk time series with another one?

I have two time series that follow a random walk behaviour. I would like to use the one I know to forecast the other one. Suppose there is a strong correlation between them with $\text{lag}=k$, for ...
1
vote
2answers
54 views

Probability of being $\$5$ up after $25$ plays of a game of Heads and Tails (fair coin) [duplicate]

In a game of heads and tails with a fair coin - you win $\$1$ if heads; lose $\$1$ if ...
3
votes
0answers
464 views

Diffusion coefficient from double-normal probability density function

The spread of individuals of species is often described by so-called dispersal kernels. The main parameter of spread is then often the variance defined as the average squared movement distance of a ...
3
votes
1answer
745 views

What's the forecast of a Random Walk with Noise model?

I have a RW with noise model defined as: $$ y_{t} = z_{t} + v_{t}$$ where $ z_{t} = z_{t-1} + e_{t}$. $v_{t}$ and $e_{t}$ are mutually independent with expectation $0$ and variance $\sigma_{v}^{2}$ ...
11
votes
2answers
13k views

What is the autocorrelation for a random walk?

Seems like it is really high, but this is counterintuitive to me. Can somebody please explain? I am very confused by this issue and would appreciate a detailed, insightful explanation. Thanks a lot in ...
0
votes
1answer
5k views

How to use ARIMA with non-stationary data

I'm in need of some help. I'm modelling daily financial data, which I have read will almost always follow a random walk model. I've confirmed this in Rusing the <...
0
votes
0answers
59 views

How to report a convergence on a random walk on Markov Chain Monte Carlo

In our psychology experiment we have found Markov Chain transition probabilities to be a good way to analyse data, and the convergence of two Markov Chains on a random walk (we used 10,000 walks) to ...
1
vote
0answers
27 views

Taking longer sample should not reduce the variance of the mean

I have not realized completely why n times longer samples should give you n times shorter variance. However, I measure performance of my programs and there are some fluctuation due to PC state ...
32
votes
5answers
27k views

Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
2
votes
0answers
37 views

Probability of losing everything in N games

Consider a gambler who starts with an initial amount of money of $£i$, obtain $£R$ with probability $p$ and lose $£J$ with probability $q=1-p$. What is the probability that it loses everything if he ...
2
votes
3answers
273 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
1
vote
0answers
44 views

Probability of a pattern in a random walk

Consider the discrete time random walk process such that $X_{t+1} = X_t + \epsilon_t$ where $X_0 = 0$ and $\epsilon_t$, is drawn from a symmetric distribution about $0$, such as the normal ...
0
votes
1answer
74 views

Modelling turnovers by a random walk. Is it right?

I need to analyse a bunch of weekly time series that reflect the turnovers of various companies. I already read that return rates or share prices show stochastic patterns that can be modelled by a ...
1
vote
1answer
627 views

Random Walk Probability Including Drift

What is the equation for the probability of a random walk with drift being equal to a specific value after n steps, given a specific standard deviation?
6
votes
2answers
3k views

Estimation of unit-root AR(1) model with OLS

Given a random walk $x_t$, $$x_t=x_{t-1}+\varepsilon_t,$$ consider estimating the slope coefficient $\beta$ in $$x_t=\beta x_{t-1}+\varepsilon_t$$ by OLS. This question and the following answer ...
10
votes
2answers
5k views

Random walk estimation with AR(1)

When I estimate a random walk with an AR(1), the coefficient is very close to 1 but always less. What is the math reason that the coefficient is not greater than one?
4
votes
1answer
4k views

Difference between random walk and process integrated of order one?

I know that an $I(1)$ process becomes stationary after differencing once. However, I somehow always equated that to its being a random walk because say having a unit root process like \begin{eqnarray} ...
0
votes
1answer
2k views

Closed-form expression for autocovariance of random walk with drift

I am working through slides hosted at Basic Time Series Models, and am not sure how to mathematically derive a closed-form expression of the autocovariance of the "random walk with drift" model. The ...
0
votes
0answers
222 views

Variance of random walk

A discrete random walk $X_t$ starts today. We are asked what to prepare for the next years. Is this correct: $ var(X_{2 years}) = 2 \: \: \int_{2\pi f}^{\pi} \:\: \phi(w) dw $ i.e. summing the ...
0
votes
1answer
904 views

How to validate random walk model

I am studying ARIMA models and find it hard to validate the model in terms of "it's a good, useful model" and "I shouldn't use that model for prediction". So at first I started with the easiest model,...
1
vote
2answers
500 views

STL + Random walk failing

We have four months of data (10 minute interval), this seems have nice pattern (at least for eye ball). We are using STL to decompose the time series and apply "random walk" to project next month ...
36
votes
8answers
10k views

Random walk on the edges of a cube

An ant is placed in a corner of a cube and cannot move. A spider starts from the opposite corner, and can move along the cube's edges in any direction $(x,y,z)$ with equal probability $1/3$. On ...
3
votes
1answer
330 views

Interpreting Auto correlation of Human walk data

My auto correlation coefficients plot of human walk looks like this. This walk data is recorded with accelerometer sensor inside the pocket. Human walk is periodic, and I need to determine that period ...
3
votes
2answers
573 views

How do I relate the std deviation of the step size, to the stdev of the endpoint of a brownian motion, if the step sizes are multiplied by a function

I know that if I take take a brownian motion of, say, 30 steps of standard deviation 1, then the standard deviation of my endpoint will be sqrt(30). But what if the standard deviation of the 30 steps ...
5
votes
2answers
1k views

Definition of random walk as a summation of independent random processes

I have a complete beginner question on random walk. As per this paper ...
1
vote
1answer
104 views

Distinguishing diffusion from white noise

I have a time series that looks like this: This comes from an experiment, and I know the following: Originally, for $t < s$ the time series is $x_t = vt + e_i$, where $v$ for this particular case ...
2
votes
0answers
230 views

Calculate 1D Random Walk Expected Iterations to return to origin

I'm trying to solve a stats problem as outlined below; I'm a bit new, however, and I'm not sure how I could solve this problem. Assume someone has lost their keys, and uses an inefficient random walk ...
1
vote
1answer
2k views

What is the Fourier Transform of a brownian motion?

I looked into this article http://en.wikipedia.org/wiki/Brownian_noise and it says that: If we have a brownian motion $W(t) = \int _{0}^{t} dW(s)$, then given that the spectral density of white noise ...
2
votes
2answers
189 views

Question on the correlation between two dependent variables

I'm working on this question and it's stumping me. Let $S_n = X_1 + \ldots + X_n$ (with $n>=1$) be a random walk with $X_1, \ldots, X_n$ be iid RV's. $$ E(X_k)=\mu,\,{\rm Var}(X_k)=\sigma^2. $...
1
vote
1answer
178 views

Is there a kind of random walk process for wich the reciprocal of the variance is a sum of reciprocals of variances of compound processes?

Context: The variance of a sum of independent random walks is a sum of their variances: $\sigma^2 = \sigma_1^2 + \sigma_2^2$. In case of a dependent random walks with bivariate normal distribution it ...
2
votes
1answer
373 views

Simplex Random Walk Mean

Hi I have two questions related to a previous question I asked here: Simplex Random Walk In this link it describes how to perform a random walk on the simplex. http://en.wikipedia.org/wiki/User:...
3
votes
0answers
161 views

Markov Chain exercise in an exam

Suppose that $X_1, X_2, X_3... $ is a Markov chain with the following transition matrix: State: | 1 2 3 --+----------- 1 |0.2 0.4 0.4 2 |0.5 0.0 0.5 3 |0.6 0.3 0.1 (forgive my attempt at a ...
1
vote
1answer
397 views

Simplex Random Walk

This link describes how to perform a random walk on the simplex using the Metropolis-Hastings algorithm: http://en.wikipedia.org/wiki/User:Skinnerd/Simplex_Point_Picking The description says: "The ...
1
vote
0answers
54 views

Random walk governed by Markov Chain

I'm trying to create an underlying markov chain whereby each state governs the probability distribution of a random walk for a stock index. Is there a more formal name for this type of process?
2
votes
2answers
134 views

Random walk under changing conditions

I have a random walk where by at certain times or conditions the increments follow one distribution, and then another distribution under different conditions - how can I model this random walk (states ...
1
vote
1answer
597 views

Simulating Diffusion/Wiener Process with Random Walk [closed]

I hope this is the right section for this kind of questions. I am trying to simulate, with MATLAB, a diffusion model starting from a Random Walk. I am using a Random Walk with information increment ...
2
votes
1answer
141 views

Random walks and Markov chains

Is it incorrect terminology to say that data follows a t-distribution random walk, for example? Does the fact the increments have this underlying distribution mean it should be referred to as a markov ...
0
votes
1answer
64 views

Financial Random Walks

Does anyone know of any good and accessible papers on the random walk modelling of financial data from a statistics perspective? Most of the papers I've found have been written by economists or ...
3
votes
2answers
597 views

Non-normal random walks

I'm aware of the simple 'proof' that shows random walks with a normal error term are non-stationary in original form and stationary in first-difference form but what happens if the errors have a ...
0
votes
1answer
189 views

Modelling a random walk with varying distributions [closed]

I have some financial data i'm trying to fit a random walk too, but the daily change increments have different distributions when studied over the last month, last year, last 5 years etc, along with ...
0
votes
2answers
88 views

Modelling non-stationary random walks

I am trying to model some data using a random walk, but after the standard increment and log (financial data) transformations for stationarity have found that, over long time frames, there is still ...
2
votes
0answers
551 views

Random walk with bivariate normal distribution

Let $X$ be a random variable from $f(x; \theta)$, where $\theta =(\theta_1,\theta_2)$. I want to: take a sample from this distribution using Metropolis Hastings algorithm and update the parameters ...
2
votes
0answers
65 views

Absorption probability in 1D RW with asymmetric step sizes, $ x<0 $

What is the probability of absorption at $ 0 $, as a function of position $ x $, for a 1D random walk (on $ \mathbb{Z} $) with asymmetric step sizes? For example, suppose that you can take two steps ...
3
votes
0answers
78 views

Detecting convergence in Random walk

I am trying to detect convergence of a random walk on a graph. After doing some preliminary research, the Geweke convergence diagnostic seems to be most commonly used for this. This diagnostic calls ...
8
votes
3answers
2k views

Random walk: kings on a chessboard

I have a question about the random walk of two kings in a 3×3 chessboard. Each king is moving randomly with equal probability on this chessboard - vertically, horizontally and diagonally. Τhe ...
3
votes
1answer
855 views

Random walk with drift in dynamic linear model

Suppose I have a dynamic linear model as defined in the dlm-package for R, see Petris 2009. $y_t = F_t θ_t + ν_t, ν_t$~$N(0,V_t)$ $θ_t = G_t θ_{t-1}+ω_t,ω_t$~$N(...
2
votes
1answer
6k views

Difference between arithmetic vs geometric random walk

I have read about arithmetic and geometric random walks. What is the difference between them?
5
votes
1answer
445 views

Using the probability generating function to find the probability of ultimate extinction

I am having problems with an exam question from a past paper, help would be appreciated: Let $ X_n $ be the number of carriers of a family name in the $n$th generation and suppose $ X_0=a $. ...
0
votes
1answer
2k views

Is non-stationary AR(p) process constant in mean?

A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?
1
vote
1answer
453 views

How to explain this unit root process?

I have a time series $X_t$ (shown below) with a structure break. The stationary test kpss.test() says it has a unit root. How to explain this? Why does $X_t$ have a ...