Questions tagged [random-walk]

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

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48 views

Is random walk with drift is random?

I see everywhere in the web that lag-plot or acf are used to see if a time serie is random. If there is no structure in the lag plot then the data are random, and if autocorrelation = 0 then data is ...
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What is the distribution of the time-to-ruin for a gambler's ruin problem that allows “pauper bets”?

In another question on this site I have derived the distribution for the time-to-ruin in the gambler's ruin problem where the wealth of the gambler follows a discrete-time random walk. In this ...
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How to invert a random walk [closed]

I have a random walk dynamic parameterized in a function let's say $f(x)$ e given a $x_0$ I can retrieve a curve of this initial value after several simulations. But I need to "invert" this ...
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Are there good examples of martingale processes that are not simple random walks?

Are there non-trivial examples of martingale processes that aren't simple random walks? I'm trying to better understand the difference between martingales and simple random walks. They look pretty ...
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Fast uniform sampling of walks from directed graph

Given a directed graph $G=(V, E)$ my goal is to sample a set of walks $W\subset\mathcal{W}$ where $\mathcal{W}$ is the set of all walks in $G$. I want each walk to be sampled with the uniform ...
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What is the distribution of time's to ruin in the gambler's ruin problem (random walk)?

In a gambler's ruin problem, where the gambler starts with a fixed amount of wealth. What is the distribution of times to ruin. That is, if each bet has a fixed payout. As I understand it, this is a ...
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MCMC: Rejecting samples outside the prior support?

I wish to implement a MCMC procedure for a posterior density which has non-trivial prior support. To clarify, this means that the parameter space has certain regions (i.e., combinations of parameters) ...
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Is CPU utilization a predictable time series? [closed]

I've been wondering whether metrics about CPU and resource utilization is a time series which can be predicted or rather a random walk which I cannot learn from. Can recognizing a pattern in the data ...
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Explain formally how expected time to hit 0 from two is the sum of the expected time to hit 1 from 2 and 0 from 1

I have a symmetric random walk on the integers with probability $p$ and $q$ of going up and down respectively started at $X_0 = 2$. Let $$ T^0 = \min\{ n > 0: X_n = 0\}, T^1 = \min\{ n > 0: X_n =...
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Appropriate distribution for simulating a random walk between two known points, with known min/max values

I have a 1-D random walker. It starts at a value of x at time t=0. It ends with a value of y ...
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37 views

Why iterations of Gibbs sampling for a bivariate Gaussian distribution can be seen as random walk?

In Section 4.4 of the excellent technical report Probabilistic Inference using Markov Chain Monte Carlo Methods, the author tries to analyze the performance of Gibbs and Metropolis algorithm with ...
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Random Walk why is $E[X_{t}] = \mu t$

Question A Random Walk can be defined as follows. $Z_t$ ($t=1,2,3,\ldots$) is a noise term with a Normal$(\mu,\sigma^2)$ distribution. Define $X_0=0$ and $$X_t = X_{t-1}+Z_t$$ for $t=1,2,3,\ldots.$ I ...
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How do we define the kernel to calculate the acceptance ratio for Metropolis-Hastings Markov Chain Monte Carlo?

I am having a lot of difficulty understanding how to apply the algorithm to a real scenario. The part that confuses me is that we are looking for a target distribution (the real distribution of our ...
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distribution of maximum random walk distance

Related to this question. Suppose I flip a fair coin $N$ times and keep track of the difference between the total number of heads and tails as I am doing it. At the end of the $N$ coin flips, I have ...
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If $y_t$ and $x_t$ are cointegrated, then are $y_t$ and $x_{t-d}$ also cointegrated?

Assume that $x_t, y_t$ are $I(1)$ series which have a common stochastic trend $u_t = u_{t-1}+e_t$. Particularly, consider the following DGP \begin{align} y_t&=\alpha_y+u_t+a_t \tag{1} \\ \end{...
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1answer
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Random walk variance in R less than expected

I am trying to prove via some Monte Carlo simulation that the variance of a random walk equals $t*σ^2$ in R. I am running the following code, 180 times, to find the variance of 180 different random ...
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Which one is more likely to be random walk?

Consider the two series in the chart below: $walkA$ and $walkB$. They are based on the same steps, although the steps come in a different order. Indeed, $stepsA$ and $stepsB$ have identical sample ...
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Can a ARIMA(0,0,0) model be stationary?

I have a time series of a stock and use its log differenced daily returns. I have conducted an ADF test for a presence of unit roots, a KPSS test as well and both confirms stationarity in the time ...
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Random walk and variance [duplicate]

If yt is pure random walk the variance Var(yt-yt-k) will (increase, decrease or remains constant) as lag k increases?
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Random walks and Pearson's correlation

One well known problem in time series analysis is spurious correlation when time series are non-stationary (and non-cointegrated). Given random walks of the form $R_i=R_{i-1}+\epsilon_i$, with $\...
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Method for predicting the probability of encountering variably-sized and distanced objects in a 2D random walk

I am working on an archaeological research problem (how people discover new resources on a landscape) and need some assistance on where to look or terms to research for a particular simulation I'm ...
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Same results for RandomWalk and Snowball sampling algorithms

I am trying to find the best sampling algorithm to obtain a sampled graph that exhibits the same distribution compared to the original graph. The metric I am using at the moment is PageRank. The ...
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1answer
83 views

Random walk with drift and trend

I am currently having a problem regarding the process, so this was the equation $$Y_t = \alpha + Y_{t-1} + \beta t + \epsilon_t$$ where, $\epsilon_t \sim WN(0, \sigma^2)$ I was calculating the $E(y)$ ...
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Probability Assigned to Distance between Two People Random Walking in a Room [closed]

In a scenario where there are two people in the rooms next to each other randomly walking in a room I want to know if we can compute PDF of distance between the two people. So the way I tried to ...
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A Skip Free Negative Random Walk

Suppose $\{X_{n}|n\geq 1\}$ is independent, identically distributed distribuited. Define $S_{0}=X_{0}=1$ and for $n\geq 1$ $$S_{n}=X_{0}+X_{1}+\cdots+X_{n}.$$ For $n\geq 1$ the distribution of $X_{n}$ ...
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what is the expected value of the dot product of two vectors

I have a little question, but I don't know that well how to answer it. I have a random walker with position vector $\vec{r} = \sum_{i=1}^N \vec{r}_i$, where i is the random walker's step. Every vector ...
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What correlation structure is necessary to ensure a random walk is almost surely bounded?

Say I have a stochastic process $\{X_t\}_{t \in \mathbb{N}}$ such that their cumulative sum $\{S_t\}_{t \in \mathbb{N}}$ is a random walk process: $$ S_t = \sum_{i = 1}^t X_i $$ If each $X_t$ is i.i.d ...
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71 views

Convergence of random walk in $R^2$ to the Brownian motion on circle

We know that the random walk generated in $R^1$ can converge weakly in distribution to the Brownian motion in $R^1$. Could anybody provide a mathematical proof, how a random walk generated in $R^2$ ...
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Random Walk in $R^2$ vs Brownian motion in $R^2$

By central limit theorem, random walk in $R^1$ converges in distribution to the Brownian motion in $R^1$. For defining a 2D random walk, is there any difference between : a) If we decompose a 2D ...
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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 $[-1,1]$....
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Variance Ratio test for 3-D random walks

The variance ratio test proposed by Lo and MacKinlay (1988) is used to detect 1-D random-walk-like-behaviour. 1-D works great for time-series data, but I'd like to adapt this test for imaging data to ...
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Test for correlated vs uncorrelated increments in random walks

Is there a test that can distinguish the strictest form of the random walk, $$P_{t}=P_{t-1}+\varepsilon_{t}, \varepsilon_{t} \sim \mathrm{IID}\left(0, \sigma^{2}\right)$$ where each step is assumed to ...
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Autocorrelation test robust to heteroskedasticity

I'm testing the random walk hypothesises 1 and 3. I'm done with the first hypothesis but am struggling with the test distribution of the third one. I'm using the autocorrelationstest. For the first ...
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Confused about stationarity and ARIMA processes

So I am quite confused about stationarity in ARIMA processes. For example, a Random Walk is an ARIMA process with order (1,0,0). Does this mean that a Random walk is stationary? Stationarity implies ...
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Is it possible to produce two [random] graphs that always pass each other

My question is whether it is possible to create two graphs that go up one point or fall one point at a time [e.g. every minute] in a random walk, and they will pass over each other for sure all the ...
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45 views

Random walking with high synthetic correlation

My question is if there is a way to create two graphs that move one point up or down at a time in a random walk, and there will be a high [synthetic] correlation between them [example: 0.8 in Pearson'...
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1answer
212 views

Metropolis-Hastings exercise with Cauchy and normal distributions [self-study]

I have the following exercise to solve and would appreciate some help. Consider a linear regression model $y = X\beta + \varepsilon$, where $y = (y_1,...,y_T)'$, $X = (x_1,...,x_T)$, $x_t$ is a single ...
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Expected number of steps for 1D circular random walk with jumps

Consider a simple 1D random walk with 50/50 probability to go left or right. The expected number of steps to reach a barrier at position $a$ or $b$ steps away is given here. If this random walk is ...
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I must solve this question on random walk, but I don't know where to start or how to do it (I need a hint)

Let $\{y_t: t=1,2,\dots \}$ follow a random walk, as in: $y_t=y_{t-1}+e_t$, with $y_0=0$. Show that Corr$(y_t,y_{t+h} )=\sqrt{t ⁄ (t+h)}$, for $t\ge 1$, $h>0$.
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What is an I(k) random walk process?

Local linear trend - I(2) process: An extension of the random walk trend is obtained by including a stochastic drift component µt+1 = µt + βt + ηt, βt+1 = βt + ζt, ζt ∼ NID(0,σ2 ζ), (3) where the ...
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Does the unconditional mean of a non stationary ARMA process exists?

Assume that we are dealing with an $ARMA(1,1)$ model: $$ y_{t} = \theta y_{t-1} + \epsilon_{t} + \alpha \epsilon_{t-1} $$ where $$ \epsilon_{t} \sim WN(0, \sigma^{2}) $$ Then, we can rewrite the model ...
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Dynamic confidence intervals for Bernoulli trails

Let $\dots, X_i, \dots\sim B(p)$ be iid. Bernoulli random variables with mean $p$, then we know from Chernoff bounds that $\Pr[\left|1-\frac{1}{np}\sum_{i=1}^n X_i\right|>\epsilon]\le2\exp(-\...
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If two random walk patterns follow each other, is it still considered a random walk?

I am wondering if these two lines, F1 and F2, representing time series, would still be considered "random walk", once the relationship between the two was discovered? Could this relationship ever even ...
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1answer
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How to implement a survival function PDF?

I'm trying to generate a set of points whose separations are given by a Lévy flight. A source that I have (Peebles 1993) says that the process goes as Starting from a [point] in space, place the ...
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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,...
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what is the difference between GCN and random walk

Anyone could explain to me what is the difference between graph convolutional network (GCN) and random walk? or they are the same? Any further explanation will be much appreciated.
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Expected HIGH and LOW of a random walk?

Given the random walk $s_n$ $$s_n=\sum x_i, \space\space\space x_i \text{ iid, }\space\space x_i \sim N[0,\sigma]$$ what are the expected highest/lowest values of the walk after $n$ steps? $$H_n=E[...
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Ornstein-Uhlenbeck process inside boundaries

I have some simulation of the Ornstein-Uhlenbeck process: ...
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1answer
65 views

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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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 + ...

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