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Questions tagged [random-walk]

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

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How would a drift diffusion model explain effects in reaction time but not accuracy?

How would a drift-diffusion model explain a case where a variable has an effect in reaction time but not in accuracy. I know that one explanation would be a speed-accuracy tradeoff, in which the ...
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62 views

random walk on Z towards the origin

Consider a random walk on $\mathbb{Z}$ with rate $a>0$ (begin no origin). The r.w. jumps one step towards the origin with probability $p$ or one step away from the origin with probability $1 −p$. ...
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64 views

Conditional probabilities for sequence of events

The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes ...
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27 views

How to calculated a probability for a sum of integers to be equal to a given value if probabilities of addends are known?

Let's assume that we have a process generating some integer numbers with different probabilities. The set of possible integers is small. For example: -2, -1, 0, 1 and 2. We also know probabilities for ...
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1answer
22 views

Are all non-stationary series random walks?

Are all (non-explosive) time series either stationary around a deterministic trend or random walks? If I run the ADF test and I can't reject the null of non-stationarity does it imply the series is ...
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32 views

Can the first difference of a time series follow a random walk?

I have a time series in levels, say $X_t$ and a variance-ratio test suggests that it does not follow a random walk. Now I differenced the time series once and the variance-ratio test suggests that the ...
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6 views

Popular stochastic model for behavior of staying at the same position?

I am looking for a popular stochastic model employed for a trajectory of a fish which tries to keep staying at the initial position against water pressure from time-varying directions. The trivial ...
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26 views

Confusing in Random Walk

I have a question about of random walk. Consider a particle starting its random walk at 0. At each step, it either moves in positive or negative direction. If the probability of moving in positive ...
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The magic money tree problem

I thought of this problem in the shower, it was inspired by investment strategies. Let's say there was a magic money tree. Every day, you can offer an amount of money to the money tree and it will ...
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39 views

Integrated Random Walk is invertible and/or stationary?

I am wondering if the integrated Random Walk $y_t$ is Stationary and/or invertible $ y_t = 2y_{t-1} - y_{t-2} + e_t $ where $ e_t$ is a White Noise. To prove non stationarity I tried to say: If, by ...
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86 views

How can we verify the intuition that in the RW-Metropolis-Hastings algorithm with Gaussian proposal too small and too large variances are bad choices

Let $d\in\mathbb N$ and consider the Random Walk Metropolis-Hastings algorithm with a Gaussian proposal kernel $Q$ such that $Q(x,\;\cdot\;)=\mathcal N_d(x,\sigma^2_dI_d)$ for all $x\in\mathbb R^d$. ...
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79 views

Average time for a random walk on the edges of a cube

In a interview i had something similar to this question Random walk on the edges of a cube. But this time there is only a ant, it takes one minute to go through a single edge. The ant can use the ...
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16 views

Random walks-heavy tailed case

Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution $P(\xi_{1}=-1)=1-C_{\beta}$ and $...
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1answer
31 views

User-Specific Activity Level Along Weeks

I am assigning Low | Medium | High activity levels to users once a week. At the end of an entire period,a user has been assign n weekly activity levels among {Low, Medium, High}. Let k be the total ...
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1answer
89 views

Probability you end up at the origin after taking $2n$ steps?

Starting at the origin on the line we take a step of unit to the left or to the right with probability $\frac12$. We do this repeatedly with independent steps. If we take $2n$ steps, what is the ...
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74 views

Expected number of steps in Gambler's ruin game with two players

Let's say we have two players A and B. Player A has 3 coins and player B has 5 coins. If player wins the other player gives one coin. During game second player probability of loosing is $2/3$, while ...
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104 views

Expectation of a random walk that can't go below zero

Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is: $$S_0 > 0$$ $$S_{i+1} = \max(S_i+x_i,\space 0)$$ $$x_i \sim N[\mu,\sigma^2]$$ Can we analytically ...
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1answer
64 views

Why is the probability of a random walk reaching 1 (in n steps) squared greater than the probability of it reaching 2 (in n steps)?

Let $S_n$ be a simple random walk. i.e. $$ S_n = \sum_{t=1}^n X_t, $$ where ${X_t}$ are i.i.d random variables with $$ X_t = \begin{cases} +1, & \textrm{w/ probability } p \\ -1, & \...
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47 views

Applying Bayesian Gaussian movement question

I have a question from my stats class that I am confused about how to proceed with. I have a general idea of what I am to do but I am not sure how to start. The question is about a car that is moving ...
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1answer
82 views

Random walk with “negative coefficient”

I was playing around in R with simulating random walks. At some point I tried this model: x = NULL x[1] = 0 for (i in 2:2000) { x[i] = -x[i-1] + rnorm(1) } ...
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Survival probability of a random walk with renewal timings

A random walker starting at time $t=0$ and location $x=0$ moves to the right ($x+1$) or the left ($x-1$). The $k^{\mathrm{th}}$ moves to the right and left occure at the times $\sum_{i=1}^{k} R_i$ and ...
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Probability of never returning to the origin until time $2n$ in asymmetric Bernoulli random walk

I have the following asymmetric random walk problem. $X_1, \cdots, X_{2n} \overset{iid}{\sim} F(p)$, where $F(p) : \begin{cases} P(X = 1) = p \\ P(X = -1) = q=1-p\end{cases}$ So what I need to ...
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14 views

What does a Drift Diffusion Model tell you about choice and reaction time?

I'm having trouble understanding what exactly the drift diffusion model, LBA, LCA, etc. models tell you about a set of 2 (or multi) alternative forced choice tasks. I know these models are supposed to ...
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0answers
34 views

How to measure “cyclicity” of a directed weighted graph?

Say you have a weighted directed graph with (potentially) some cycles in it. You want to have some sort of a measure of how "cyclical" this graph is. The requirements are: This measure C=0 on an ...
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78 views

The Hessian of multinomial Probit model

I wanted to implement multinomial probit in Bayesian with random-walk Metropolis Hasting. To achieve the best numerical efficiency when drawing $\beta$, I need to use the hessian matrix of $\beta$. ...
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19 views

Modelling product purchase history as a random walk in n-space

I have a large dataset of customers making monthly purchases of multiple products. Customers usually purchase between 3 and 10 products, from a large product list (1000s). I'm interested in clustering ...
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1answer
25 views

Difference between $2^{nd}$ order random walk and personalized pagerank

I've been recently working with graph sampling, and I can't seem to find useful explanation of the following two aspects. On one side there are pagerank-based algorithms, which converge to a ...
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1answer
106 views

Proving that a random walk that diverges to infinity may not become negative

Consider a random walk $S_n= \sum_{k=1}^n X_k$, where $\{X_k\}_{k=1}^\infty$ are independent and identically distributed random variables. Assume that $S_n \rightarrow \infty$ almost surely as $n \...
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1answer
171 views

Proving that a random walk using a maximum likelihood estimator can diverge to infinity

Consider a sequence of continuous random variables $\{X_n\}^\infty_{n=1}$ that are independent and identically distributed under the probability density function $f_\theta (x)$, where $\theta \in [\...
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462 views

Are S&P 500 monthly (or annual) returns a random walk?

I'm using financial software that assumes that yearly market returns are random and independent in their Monte Carlo analysis. Its not clear to me that this is the case. Is there an easy way for a "...
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1answer
178 views

Expectation of the absolute value in a sequence of Bernoulli trials

On this tweet: Can I get some help in understanding the proposed solution by N. Taleb: It is not clear how he describes success, i.e. $n-x$ to come up with $\binom{n}{n-x}.$ It almost seems as ...
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334 views

random walk and covariance stationary

I was preparing for CFA and encountered this question, which is quite puzzling. To use autoregressive model, it has to be covariance stationary (same mean, covariance). If a model's residual is not ...
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Why switching edges definition changes the result of random walk search for communities? (walktrap)

If a graph object is a not directed graph, then the following set of operations should yield the same result: ...
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1answer
428 views

Showing that R-squared might not be useful in time series data

I understand that using $R^2$ in time series models may not be the best as $R^2$ is non-decreasing. I also read this post: What is the problem with using R-squared in time series models? on the ...
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1answer
113 views

Principal Components of Random Walk

In this blog figure 4 shows that the principal components of a random walk are sinusoidal with increasing frequency for decreasing eigenvalue. Is there an intuitive way of understanding this? If I ...
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43 views

What is the “distance” of a random walk on a graph?

What is the definition of the distance of a random walk? In the Statistical significance of a cluster of this paper, and this ...
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883 views

How to simulate Lévy flights?

I found this code: ...
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51 views

Brownian bridge to unknown via extremum

Suppose, I know what's the minimum $\min$ of a random walk $w_t$ in period $[0,\Delta t]$. I also know $w_0$ and $\sigma$. How to construct the Brownian bridge for the latter period? I guess it's not ...
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149 views

Estimate standard deviation of random-walk using Kalman filter

I'm new to Kalman filters so this might be a stupid question. I created a Kalman filter that takes in time series observations and estimates the mean of that time series. This is simply modeling a ...
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1answer
116 views

Random Walk Process - Time Series

Is it true that the mean of a random walk process does not depend on time and the sequence can be considered mean stationary?
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1answer
202 views

Power Spectral Density of Random Walk

The Brownian motion has a power spectral density (PSD) dependency on frequency like $\frac{1}{f^2}$. As far as I understand, power spectral density is defined only for wide sense stationary processes ...
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150 views

Variance sum of two independent random walks

I have two random walks, which represent fishing mortality in season 1 and season 2 of year $t$ ($X_{t,\mathrm{summer}}$ and $X_{t,\mathrm{winter}}$). If I add up both series to obtain annual fishing ...
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1answer
205 views

Does it make sense to have the dependent variable in growth rates or rather in levels?

I am currently investigating the impact of uncertainty on investment dynamics. I have an unbalanced panel data set (approximately 100,000 observations). To study the relation between investment and ...
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2answers
61 views

Forecasting average values with varying number of observations

Every day $t$ we observe several (independent) realizations of a variable $X_t$. This variable is the sum of a time dependent mean value and white noise: $$X_t=\mu_t+\epsilon_t$$ You can assume $\...
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2answers
316 views

How to prove that the probability of spurious correlation increases with random walk length?

Define a simple random walk $y_{t}$ as: $$y_{t} = y_{t-1} + 2\times Bernoulli\left(0.5\right)-1,$$ so that at time $t$ the value of $y$ equals its previous value plus a perturbation from the "flip-a-...
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1answer
325 views

Interpretation of an I(2) process?

I know that an ARIMA(0,0,0) process is white noise and ARIMA(0,1,0) is a random walk, Is there an interpretation of what an ARIMA(0,2,0) process is?
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1answer
158 views

Predictor for averaged Brownian motion

The best forecast (predictor) for a Brownian motion at time $t+h$ is the present value at time $t$ since it's a martingale. The same holds for random walks with independent steps and without drift. ...
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2answers
4k views

How to interpret ARIMA(0,1,0)?

I ran the auto.arima()command in R on a set of data and it chose the appropriate model to be ARIMA(0,1,0). I know ARIMA(0,0,0) is just white noise, but what does ...
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103 views

Random walk touching or exceeding thresholds

What is the formula to estimate the probability of a random walk touching or exceeding a particular threshold? The threshold starts and stops at particular times. (Without starting and stopping times ...
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387 views

Stochastic or Deterministic Trend: Supported by the Augmented Dickey-Fuller Test

Below are the sequential steps/question regarding my problem: I am attempting to specify a VAR model in order to analyze impulse response functions. In plotting my first variable (Figure 1) I ...