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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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Conditional variance of random walk with given start points and ending points

If I have a random walk with 0 drift and I observe that $X_1 = x_1$ and $X_k = x_k$, bu I don't have all the points from $X_i$ for $i \in \{2, ..., k-1 \}$, how do I estimate them and given an CI? I ...
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How to guess a random walk to achieve max sample correlation?

Define this 1-D discrete random walk start from 0: roll a die (the die may or may not be fair, the fixed probability of each face is unknown to the observer/guessor)...
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Why is a coefficient in front of $Y_{t-1}$ in a random walk (equal to 1) not an autocorrelation coefficient? [duplicate]

As it is said everywhere autocorrelation measures the correlation between a time series variable and its lagged values at different time intervals. Then why can't we say that coefficient in front of $...
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How can I simulate the path between two defined points but also define the overall step variance?

As stated in the question. I’m wondering if it’s possible to simulate a random walk between two fixed points (always start at A and finish at B) where the variance of the difference of steps is also ...
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Non-stationarity and panel data

Spurious regression happens when we regress two independent non-stationary time-series: $$y_t = y_{t-1} + u_t \qquad x_t = x_{t-1} + v_t \qquad \operatorname{corr}(y_t, x_t) = 0$$ Then if we do the ...
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How to perform random walk on multilayer network to predict new edges

I have a multi-layer network that is a union of 3 networks (field of human biology/ Omics data). The 3 networks have dense connections within each other (local), however sparse connections to each ...
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experimenting random price movement

Random price movement Consider the following: The price of an apple starts at 1 dollar. On each day, the price will change -10% or +10%, with equal probability. You buy this apple on day 1, and sell ...
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Checking if RW() = ARIMA(0,1,0) with drift

I am trying to confirm the following statement that in R fable package: ...
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2 answers
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Gridsearch on ARIMA favours random walk

I am working on a time-series forecasting problem with ARIMA. Since long-term predictions were not good, I've started using a "rolling ARIMA" like explained here ...
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Should I consider it white noise?

I have an hourly time series that I want to forecast. Prior to modelling it, I tested it for random walk. The ACF and PACF plots for the time series are as follows: Since the PACF has a high value at ...
Om Mali's user avatar
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5 votes
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Divergence of $e_{i+1}\leftarrow e_i - x_i e_i$ for Cauchy $x_i$

Suppose $e_0=1$ and $e_k$ evolves according to the following recurrence with $x_i\sim \operatorname{Cauchy}$, IID draws from standard Cauchy random variable. $$e_{i+1}\leftarrow e_i - a (x_i e_i)$$ ...
Yaroslav Bulatov's user avatar
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A Pólya urn random walk with step probability $p$ and an absorbing state. Is there for this random walk a $p>0.5$ where absorption is almost certain?

This question relates to: How to pick the winner in the "Play the Winner" treatment assignment scheme (Urn model) which is like a Pólya urn model where the additions of balls into the urn ...
Sextus Empiricus's user avatar
2 votes
3 answers
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Interview Question: What is the probability they will be home in more than 30 minutes?

The following is an interview question: A student leaves Univeristy (U) to walk Home (H). It is a distance of 4 blocks in a straight line. At each crossing, they toss a coin deciding whether to move ...
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Why isn't Random walk with trend non-stationary according to ADF?

A random walk with trend doesn't have unit root. So, null hypothesis will be rejected. Hence, according to alternative hypothesis, since it doesn't have unit root, it will become stationary process as ...
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Cross sectional variance of random walks

Suppose that there are $J$ markets, and prices in every market $j$ follow a random walk. That is, for any time $t$, the price $p_t$ is the sum of shocks up to $t$: $$p_t = \sum_{i=1}^t \epsilon_i^j$$ ...
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Random walk on cube with condition of not returning

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 average, ...
Charlie's user avatar
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Is a random walk i.i.d.? [closed]

Assuming that we have an AR(1) process: $$X_t=\rho X_{t-1}+\varepsilon_t,\quad\varepsilon\sim IIDN(0,1)$$ And further assuming that $\rho=1$ and $X_0=0$ we have a random walk process: $$X_t=\sum_{i=1}^...
Rstrobaek's user avatar
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Random walks with momentum

I am interested in random walks with momentum in 1D, ie., random walks in which there is a certain probability $p$ to take a step in the same direction as the last step, and a probability $q = 1-p$ to ...
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Deriving parameters of gaussian noise that will lead to bounded random walk

Given a bound on random walk, I am trying to derive the parameters of a normal distribution of noise, which when added to a signal and integrated will lead to random walk within specified bounds. My ...
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Random Walk Question - How to determine faster method

A man had too much to drink and is standing on a bridge that is 100ft long. He’s currently at the 15th foot, but has a tendency to either move forward or backward one meter with a 50% chance for each ...
Rohan's user avatar
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The Dead Drunk Man

I came across the following question in Fifty Challenging Problems in Probability with Solutions by Frederick Mosteller in page no. $51:$ From where he stands, one step toward the cliff would send ...
Anwesh saha's user avatar
17 votes
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What is the expected number of children until having the same number of girls and boys?

A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/...
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Show that $\xi _{n}^2-n$ is a martingale

Let $\xi_{n}$ be a symmetric random walk, i.e, $\xi_{n}=\eta_{1}+\eta_{2}+\ldots+\eta_{n}$ where $\eta_{1},\ldots$ is a sequence of independent identically distributed random variables such that $P\{\...
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Just passing through: Estimating the probability that a changing quantity will pass through some value within a time interval?

Suppose I have a quantity that is measured periodically, that is continuous in the following sense: In order for the quantity x to move from x=a to x=b, x must pass through all the values between a ...
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Rejection of ADF-test for log returns and AIC selected ARIMA(0,0,0) and ARIMA (0,0,0) with a drift?

I use monthly log returns for some stock portfolios and rejects the null of the ADF-test for both. Hereafter I use AIC to select best fitting models using auto.arima in R. The selected models are ...
NotJohnLeCarre's user avatar
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Do 1D random walk probability distributions generally converge to gaussian functions?

Suppose I have a 1D random walk, where the step size is not a constant but a random variable that is a function of a set of independent continuous random variables $x^i$ with given probability ...
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Intuition behind occurence of non central chi squared distribution in conditional coordinates of a random walk

Description of background Consider a 2d random walk with drift: $$X(t) = \sum_{k=1}^t X_k \\ Y(t) = \sum_{k=1}^t Y_k$$ where each $X_k$ and $Y_k$ are independently exponentially distributed with rate ...
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Overlapping vs non-overapping windows in random walk

Consider a simple random walk. I am trying to compute the variance of differences over a window of certain size dw which could, for example, model returns of a ...
Botond's user avatar
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Calculate mean and autocovariance for random walk to check stationarity

I am confused by the second line and the third line of the autocovariance calculation. Like how the var() and the 0 come, and why there is t*sigma^2 at the end.
Vivien's user avatar
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1 vote
1 answer
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Understanding the autocorrelation of random walk

I have been trying to derive the autocorellation for a random walk. I obtained the following result: $\rho_k(t)=\frac{\operatorname{Cov}\left(x_t, x_{t+k}\right)}{\sqrt{\operatorname{Var}\left(x_t\...
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4 votes
2 answers
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How are group sequential analysis, random walks, and Brownian motion related?

Assume that I am planning a clinical trial comparing two groups using a binary outcome. I will do the $\chi^2$ test after 3 equal enrollment intervals: interim test #1 after $m_1$ enrollments in ...
Mkanders's user avatar
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1 answer
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Correlation with a random walk

Correlation with a random walk: Thanks for kind explanations. But I am still confusing. A random walk repeats previous values plus stochastic fluctuations. Then, can exogenous factors influence a ...
Youseop Shin's user avatar
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1 answer
726 views

Is random walk stationary or non-stationary?

I'm confused with the concept "stationarity". Reference: https://otexts.com/fpp3/stationarity.html I believe random walk is non-stationary and change (difference) in random walk is ...
dmjy's user avatar
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How many "effective" observations from sliding window sampling?

I have data on the trajectory of a particle and am interested in the displacement of the particle in a given time period. In particular, I have information on how much a particle has moved in each ...
arm61's user avatar
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1 vote
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Highly persistent exchange rate within a bound: alternatives to integration and stationarity tests

I was trying to assess stability using a stationarity test. If an exchange rate is fix, I thought, it must be stationary around its parity. Unfortunately I realised this approach does not work. Hong ...
Khairon's user avatar
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2 votes
1 answer
218 views

Probability that a simple 1d random walk is between [-k,k] in 100 moves

What is the probability that a simple 1d random walk is between (-k,k), exclusive, in 100 moves? My initial though was: $1-\sum_{i=k}^{100}P_i$, where $P_i$ stands for the probability that it reaches $...
user334639's user avatar
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Random walk and Poisson process

(1) A point is chosen at random in a circle with center at the origin and radius R. That point is taken as the center of a circle with radius X where X is a random variable having density f. Find the ...
Win_odd Dhamnekar's user avatar
1 vote
0 answers
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What is the weight of walk and the intuition behind Prune and Enrichment sampling?

For context, I am reading this document: https://webspace.maths.qmul.ac.uk/t.prellberg/papers/pub084pre.pdf. I am on the section of pruned and enrichment sampling of a random walk. The document neatly ...
megamence's user avatar
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1 answer
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How to use Stirling's formula of n! in this probability computations in random walk?

I want to compute $ \binom{2n}{n} p^n (1-p)^n = \frac{(2n)!}{n!n!}(p(1-p))^n, n=1,2,3...$ By using an approximation, due to Stirling, which asserts $ n! \sim n^{(n +\frac12)}e^{-n}\sqrt{2\pi}$ Where ...
Win_odd Dhamnekar's user avatar
1 vote
2 answers
622 views

Properties of cumulative sum of a random walk?

A random walk can be generated by computing the cumulative sum of a list of random numbers. ...
asmaier's user avatar
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Stationarity of Random Walk (Backshift Operator)

I have a question regarding the backshift operator. A random walk $X_{t} = X_{t-1} + \epsilon_{t}$ can be rewritten as $(1-B)X_{t} = \epsilon_{t}$. We know that the first difference of a random walk ...
DLTS's user avatar
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1 vote
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Random Walks Question [closed]

I am trying to solve this question by using the reflection principles. Let a>c>0 and b>0. A is the set of all paths of a random walk which end at c in their final n’th step. B is the set of ...
Jack Wills's user avatar
1 vote
0 answers
18 views

Non anticipative sampling an ARIMA(1,1,0) process with known terminal value

I have an $\mathrm{ARIMA}(1,1,0)$ process $X_t$, for which I know the values $X_0=a$ and $X_T=b$. I want to sample paths $(X_t)_{t=1..(T-1)}$ consistent with the boundary conditions. One way to do it ...
AndreA's user avatar
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1 vote
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When are continuous-time models important?

In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ...
Carl's user avatar
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1 vote
1 answer
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What type of Markov Chain is a random walk of a Knight on a chessboard?

Assume we have the following chessboard and we have a knight that starts at the top left corner of the board. On every move the Knight chooses reachable square (i.e. a valid chess move a Knight can ...
Gooby's user avatar
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2 answers
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Why can de-noising diffusion models be sampled with Gaussian distributions?

In de-noising diffusion models 1 the latent is typically sampled with a unit normal distribution, and then the sample (e.g. image) is generated by iteratively removing noise during the backwards ...
Andy's user avatar
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1 vote
0 answers
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lowerbounding the expectation of maximum of $K$ random walking

Let $X_{i,j}$ be $K \times N$ i.i.d. random variables such that $P(X_{i,j} = 1) = P(X_{i,j} = -1) = \frac{1}{2}$, and $S_p = \sum_{q=1}^{N} X_{p,q}$ be $K$ i.i.d. random variables, each of which is an ...
Ruiyuan Huang's user avatar
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Unbiased random walk : why is a random sample not calibrated

To simplify, consider unbiased random walks with absorbing barriers at 0 and 100. A random walk starting at X has an expected probability to hit the barrier 100 of exactly X%. However, it seems that ...
B. Decoster's user avatar
4 votes
1 answer
709 views

Time-Series: Testing for stationarity and random walks

My goal is to test the weak-form efficient market hypothesis using time-series on prices of various stocks listed on S&P 500. According to theory, a particular stock is said to be weak-form ...
Hatori_Hanzo's user avatar
6 votes
1 answer
2k views

3-D random walk: average distance after N steps

I am calculating the average distance in a 3-D random walk process after N steps. Each step is one unit long and the angle is randomly distributed around the origin. After N steps, what is the average ...
shaw shen's user avatar

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