Questions tagged [random-walk]

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

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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 ...
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 ...
31
votes
2answers
4k views

Why are random walks intercorrelated?

I have observed that, on average, the absolute value of Pearson correlation coefficient is a constant close to 0.560.42 for any ...
23
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1answer
3k views

Hamiltonian Monte Carlo vs. Sequential Monte Carlo

I am trying to get a feel for the relative merits and drawbacks, as well as different application domains of these two MCMC schemes. When would you use which and why? When might one fail but the ...
20
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4answers
1k views

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 ...
18
votes
1answer
9k views

MCMC on a bounded parameter space?

I am trying to apply MCMC on a problem, but my priors(in my case they are $\alpha\in[0,1],\beta\in[0,1]$)) are restricted to an area? Can I use normal MCMC and ignore the samples that fall outside of ...
18
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2answers
1k views

Random walk with momentum

Consider an integer random walk starting at 0 with the following conditions: The first step is plus or minus 1, with equal probability. Every future step is: 60% likely to be in the same direction ...
15
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2answers
3k views

What does it mean to say that an event “happens eventually”?

Consider a 1 dimensional random walk on the integers $\mathbb{Z}$ with initial state $x\in\mathbb{Z}$: \begin{equation} S_n=x+\sum^n_{i=1}\xi_i \end{equation} where the increments $\xi_i$ are I.I.D ...
11
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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 ...
10
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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?
10
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2answers
18k views

Why is a random walk not a stationary process? [duplicate]

In the book Analysis of Financial Time Series by Rue Tsay, I read: A time series $\{p_t\}$ is a random walk if it satisfies $p_t = p_{t−1} + a_t$ where $p_0$ is a real number denoting the ...
10
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1answer
507 views

Why is an unbiased random walk non-ergodic?

Wikipedia says "An unbiased random walk is non-ergodic." Let's look at a simple random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1$ or $−1,...
9
votes
2answers
934 views

Computing the cumulative distribution of max drawdown of random walk with drift

I am interested in the distribution of the maximum drawdown of a random walk: Let $X_0 = 0, X_{i+1} = X_i + Y_{i+1}$ where $Y_i \sim \mathcal{N}(\mu,1)$. The maximum drawdown after $n$ periods is $\...
8
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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 ...
7
votes
2answers
9k 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 ...
7
votes
2answers
534 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-...
7
votes
1answer
123 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 $x\in\mathbb R^d$. ...
6
votes
2answers
1k views

Prove that a simple random walk is a martingale

Note that $a$ has a mean of 0. My approach: $$X_t=X_{t-1}+a_t$$ $$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + ...
6
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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 ...
6
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1answer
211 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 ...
5
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1answer
32k views

Explain what is meant by a deterministic and stochastic trend in relation to the following time series process? [closed]

Explain what is meant by a deterministic and stochastic trend in relation to the following time series process? $y_t = c + y_{t-1} + \varepsilon_t$ where $\varepsilon_t\sim iid(0, \sigma^2)$ this ...
5
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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 ...
5
votes
1answer
339 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 \...
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 $. ...
5
votes
0answers
136 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 ...
5
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0answers
460 views

Tuning MALA (Metropolis-adjusted Langevin) proposal

I'd like to implement a version of Metropolis-adjusted Langevin sampling, but I'm unsure how to go about tuning the parameters of the proposal density. My understanding is that in MALA, a proposal ...
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} ...
4
votes
2answers
434 views

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? ...
4
votes
2answers
2k 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 ...
4
votes
3answers
159 views

Simulating random walk with “known” prediction

Suppose a random walk that looks a bit like this set.seed(420) x=rnorm(1000) y=rep(NA,length(x)) y[1]=x[1] for (i in 2:length(x)) { y[i]=y[i-1]+x[i]*0.7 } But ...
4
votes
1answer
440 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
1answer
77 views

Variance of a 2D random walk

let define a 2D random walk by $$ \sum_i A_i X_i $$ where $A=[\cos(\theta)\ \sin(\theta)]^T$, $\theta$ is a random variable in the range $[0,2\pi]$ and $X$ is a scalar random variable between $[-1,1]$....
4
votes
1answer
588 views

Random Walk Metropolis Hastings implementation in R using log scale

Context I looked literally everywhere but I couldn't find a complete implementation of the Random Walk Metropolis-Hastings algorithm using the log scale. By log scale I mean that we are working with ...
4
votes
0answers
172 views

Pareto optimality in Metropolis sampling

In the Metropolis sampling algorithm, we have some function $f(x)$ proportional to a probability distribution $P(x)$. To generate a random walk with stationary distribution $P(x)$, we generate a ...
4
votes
0answers
75 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 ...
4
votes
0answers
52 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 ...
4
votes
0answers
420 views

Distribution of high and low of a random walk

If I have a one-dimensional random walk with position $X(\text{t})$ with $X(0)=0$ and $\text{Var}(X(1)) = 1$, and I observe $X(1) = \text{c}$, what are the distributions of the minimum and maximum ...
4
votes
0answers
68 views

Random walk with restricted graph knowledge

I have a very large graph and a function of its vertices, and want to estimate mean value of this function. It's not possible to sample vertices uniformly in this problem, so a reasonable choice for ...
3
votes
2answers
574 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 ...
3
votes
2answers
599 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 ...
3
votes
2answers
190 views

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 ...
3
votes
1answer
667 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
203 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. ...
3
votes
1answer
4k views

Random Walk with Drift: Why is the change in a trending variable also a function of a random variable when $E(\epsilon_t) = 0$?

I came across Pearson’s companion site of Murray, M. P. (2005). Econometrics: A modern introduction. Pearson Higher Education. While skimming through the related lecture slides here http://wps.aw.com/...
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(...
3
votes
2answers
37 views

Variance of Marginals of Continuous Random Walk Conditioned on Future Value

Consider the $N$ i.i.d. values $$ X_i \sim \mathcal{N}(0, \sigma^2) $$ such that $$ Z_i = \sum_{j=1}^i X_j $$ I am interested in the distribution $$ f(X_i | Z_N = z) $$ Mean Under the condition,...
3
votes
2answers
297 views

Random Walk Process in 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?
3
votes
1answer
814 views

Estimating Covariance Matrix of Innovations of Multivariate Random Walk

Suppose that I have a multivariate random walk: $X_{t+1} = X_t + \epsilon_t$ where $\epsilon_t \sim N(0,\Sigma)$ Estimating the covariance matrix $\Sigma$ is straightforward from first differences $...
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 $\sigma_{v}^{2}$ ...
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 ...

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