# Questions tagged [random-walk]

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

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### How can I prove the simple random walk is a Markov process?

I know a simple random walk is defined as $X_t=X_{t-1}+w_t$, but how can I modify this equation is show it is a Markov process?
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### Can an random walk ARIMA model have a nonzero constant term?

From what I'm reading it seems like a nonstationary ARIMA model can have a nonzero constant term. I'm not understanding how this can happen. Suppose we have an AR(1) model where $\phi_1=1$. If p is ...
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### Probability of Normal (Gaussian) random walk crossing threshold within k steps

Let $x[n]$ be a Gaussian random walk, so $x = 0$ and $x[n+1] = x[n] + v$, where $v$ is an independent random variable with normal distribution, $0$ mean and standard deviation $s$. What is the ...
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### Random walk on the edges of a square

A bug is at one corner of a square. What's the expectation of the number of steps it takes, to reach the opposite corner? Each step takes it to an adjacent corner, with either corner equally likely.
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### Confused about how Random Walk Metropolis algorithm work?

The following picture explains how the Random Walk Metropolis algorithm walk throughout time from $t=1$ to $t=99$. At times $t=1$ and $t=2$, things are fine to understand, somehow, I am lost ...
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### Which come first? (random walks)

Suppose I have a continuous time random walk in one (non-time) dimension, based on not-necessarily-Gaussian white noise. I know its value at the beginning and end of an interval, and from inside of ...
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### Simple Random Walk question

I am having trouble with these questions, I understand the rules I am meant to use, such as Markov property and independent increments yet I am having issues applying that to the question. I am also ...
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### Random-walk prior with ridge-like regularizarion?

I am working with a model that contains a large number of coefficients, arranged in an ordered vector $\beta_1, \dots, \, \beta_N$. I have some prior knowledge that could be used to improve the ...
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### Random walk 1-D with fixed number of steps and distance

I'm trying to program a random walk in one dimension with a fixed number of steps, which lenght is a real number picked from a distribution to be specified (in particular a polynomial one), and a ...
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### Spurious Regressions (Random Walk)

I have learned that the regression of a random walk process on another leads to seemingly statistically significant relationships, if you just use OLS. However, why do we get such large t-statistics? ...
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### Matlab Regenerating figures: Simulating Brownian Motion via Random Walks

I'm trying to understand the relation between discrete-time random walk process and continuous-time wiener process. I'm reading this lectures and to understand concepts and proofs I need to ...
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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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### Dickey-Fuller test interpretation (urca package)

I am having trouble with interpreting the Dickey-Fuller test on a time series using the ur.df() function in the urca package. I already read this thread but still need some advise. The command is: ...
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### 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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### Calculating Conditional Expected Value in R

I have a stock whose returns follows a Random Walk with mu= 6% and sigma= 20% I would like to calculate the 10 returns of this distribution that I would get ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $x\in\mathbb R^d$. ...