Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [random-walk]

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

0
votes
0answers
18 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 ...
1
vote
0answers
13 views

Analyzing animal occurrence data to detect a movement bias in R

I have occurrence data for animal locations over several months. These UTM coordinates are coupled with a date and time of occurrence, but they differ from tracking data in that it is not over ...
6
votes
1answer
71 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 $x\in\mathbb R^d$. ...
1
vote
1answer
73 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 ...
0
votes
0answers
15 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 $...
2
votes
1answer
29 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 ...
2
votes
1answer
76 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 ...
1
vote
0answers
42 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 ...
5
votes
0answers
99 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 ...
1
vote
1answer
61 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, & \...
0
votes
1answer
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 ...
0
votes
1answer
70 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) } ...
4
votes
0answers
53 views

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 ...
2
votes
0answers
25 views

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 ...
0
votes
0answers
12 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 ...
2
votes
0answers
29 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 ...
0
votes
0answers
66 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 $\beta$. ...
0
votes
0answers
18 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 ...
0
votes
1answer
20 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 ...
4
votes
1answer
86 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 \...
0
votes
1answer
139 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 [\...
0
votes
0answers
18 views

How does the range of a random walk vary with sample size?

I want to compare the ranges of 2 sets of data, but one set has N times the number of steps as the other. I'm willing to treat the underlying process as a random walk. Is sqrt(N) an appropriate ...
0
votes
2answers
396 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 "...
2
votes
1answer
140 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 ...
0
votes
1answer
287 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 ...
1
vote
0answers
14 views

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: ...
1
vote
1answer
344 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 ...
6
votes
1answer
103 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 ...
0
votes
0answers
40 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 ...
1
vote
0answers
694 views

How to simulate Lévy flights?

I found this code: ...
4
votes
0answers
49 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 ...
1
vote
0answers
140 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 ...
0
votes
1answer
106 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?
0
votes
1answer
160 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 ...
0
votes
0answers
139 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 ...
2
votes
1answer
166 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 ...
1
vote
2answers
58 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 $\...
4
votes
2answers
295 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-...
3
votes
1answer
233 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?
3
votes
1answer
153 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. ...
6
votes
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 ...
1
vote
0answers
91 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 ...
3
votes
0answers
327 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 ...
0
votes
1answer
75 views

How to add random walk in rstanarm [closed]

I have used rstanarm GLM model without the intercept like below in R ...
1
vote
1answer
271 views

Variance of random walk, mean reverting and trending series

I am reading Ernie Chan's blog post "Mean reversion, momentum, and volatility term structure". It says that To be precise, if $z$ is the log price, then volatility, sampled at intervals of $\tau$, ...
0
votes
0answers
82 views

Ito's integral formula for non-standard Brownian motion

Concerning Ito's integral formula, $$\int_0^t B(s)dB(s) = \frac{1}{2}B^2(t)-\frac{1}{2}t,$$ the MIT lecture notes give a proof that "the standard Brownian motion has a.s. finite quadratic variation ...
4
votes
1answer
265 views

How to implement a uniform random walk on a simplex?

I am looking for an uniform random walk algorithm on a simplex for MCMC purposes. Hence the process should on average spent the same time in any given area. I want this to be my proposal algorithm. ...
4
votes
2answers
1k views

Moving Average, Exponential Smoothing, and Random Walk for Forecasting

I would like to confirm my understanding. Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk ...
2
votes
1answer
291 views

Runs test and Durbin-Watson test yield different outcomes

I have analysed the market return using the runs test and the Durbin-Watson test to determine whether the return series follow the random walk or not. The problem I have found is that some return ...
0
votes
1answer
71 views

Proof the increment variance of the Scaled Random Walk

Start by defining the Symmetric Random Walk: $$ M_t = \sum_{i=1}^{t}X_i, ~~ \text{with}~X_0=0 $$ where $X_i$ is equal to 1 or -1 with $p=(1-p)=0.5$. Consider $t > s$, the variance of its ...