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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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18 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 1/2. We do this repeatedly with independent steps. If we take 2n steps, what is the probability ...
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25 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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91 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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57 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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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 ...
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1answer
64 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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51 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 ...
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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 ...
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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 ...
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0answers
27 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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44 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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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 ...
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1answer
17 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
73 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
126 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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33 views

Finding stable distribution in Markov chain using random walk

I have heard that there are random walk algorithms for approximating the stable distribution in a Markov chain. However, I have not been able to find more information on the internet. Could someone ...
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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 ...
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2answers
349 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
113 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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0answers
67 views

Intuitive explanation of GSVA analysis

I'm trying to understand the way the GSVA analysis is working behind the scenes.And I was wondering if there is any way to understand it more intuitively the whole process. So at first according to ...
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1answer
249 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
289 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
88 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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32 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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567 views

How to simulate Lévy flights?

I found this code: ...
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0answers
46 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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130 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
94 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
147 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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130 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
146 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
57 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
268 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
171 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
150 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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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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76 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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297 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 ...
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1answer
66 views

How to add random walk in rstanarm [closed]

I have used rstanarm GLM model without the intercept like below in R ...
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1answer
230 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 $\tau$, ...
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78 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 ...
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1answer
244 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. ...
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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
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1answer
253 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 ...
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1answer
58 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 ...
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169 views

Is a random walk necessarily a martingale?

I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale." Although it seems logical with the used ...
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1answer
65 views

Estimating error from an autocorrelated random walk

Hello Cross Validated, I am trying to simulate autocorrelated, lognormal values such that their SD matches that of an existing time series. I plan to generate these values using the following ...
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1answer
219 views

Error term in Random walk with drift

In random walk model the error term is generally modelled using normal distribution. I have two questions. Q1. How to calculate the parameters of error term in case of Random walk with drift. Q2. ...
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50 views

the length of a random walk in two dimensions

This is a matlab code that could generate a random walk in two dimensions of length 1000: xy = cumsum (-1+2*rand(1000,2),1). The 'stepsize' is between $[-1,1]$. In this case, we are talking about ...