Questions tagged [random-walk]

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

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30 views

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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42 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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41 views

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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48 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 $[-1,1]$....
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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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19 views

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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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
71 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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66 views

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
47 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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181 views

How to make random walk stationary?

I know that random walk is modeled as; $Y_t = Y_{t-1}+ u_t$ ; It is known that it is not a stationary model since the variance of observations at different times are not same. As a general, remedy ...
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296 views

Random walk based on coin toss [closed]

You are in an open field with two coins and you alternate flipping the coins and taking steps by the following rules: ...
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512 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 ...
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Use of the inversion method in sequential sampling to “invert” a random walk

Let $M\subseteq\mathbb R^3$ be Borel measurable, $\lambda$ be a $\sigma$-finite measure on $\mathcal B(M)$, $k\in\mathbb N$, $I:=\{0,\ldots,k\}$, $q$ be a probability density on $\left(E^I,{\mathcal E}...
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1answer
63 views

Mean and variance of a Random Walk with lower boundary

Consider a random walk $S_n$ that starts at $S_0$ and terminates if it hits a lower boundary of $0$. $$ S_n=\begin{cases} 0, & \text{if } S_{n-1}=0\\ 0, & \text{if } S_{n-1}+x_n\le0\\ S_{n-...
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1answer
48 views

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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85 views

Probability of Normal (Gaussian) random walk crossing threshold within k steps

Let $x[n]$ be a Gaussian random walk, so $x[0] = 0$ 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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How to calculate average for financial random walk?

Let's look at a simple financial random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1\$$ or $−1\$$ with a 50% probability for either value, ...
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477 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,...
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1answer
51 views

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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1answer
53 views

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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1answer
139 views

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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1answer
33 views

Random Walk Stopping Time Calculations

Let $S_n$ be a random walk with $P(S_{n+1}=S_n+1|S_n)=p<\frac{1}{2}$ and $1-p=q=P(S_{n+1}=S_n-1|S_n)$. Let $\tau=min(n:S_n=0)$ How may we show that for any positive integer $x,\mathbb{E}[\tau|...
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162 views

Superposition of random walk and autoregressive process

Let us consider the following model: $$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(0, \sigma^{2}...
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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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mean time to return random walk continuous whit drift to origen

Let $S_t$ a random walk with rate $a>0$ on $Z$ that such has a drift in direction of $0$, this is, if $(0,0)\in Z\times[0,\infty)$, and defined a sequence $(S_n, T_n)_{n\in N}$ such that $(S_0, T_0)...
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Any ideas on how to forecast this timeseries?

I have the following timeseries and unfortunately no other information except time and holidays. What methods could work for a probabilistic forecast of this? I thought about some regime changing ...
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155 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 ...
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73 views

Proof for how the drift estimator, for a random walk with drift, is unbiased?

Random walk with drift formula is: (Yt = α + Yt-1 + εt ) How do I go about checking that the drift estimator α-hat is unbiased.. which is proving that E(α-hat) = α? Is this something I would need ...
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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 ...
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How to apply the diffusion maps when the matrix is PSD but not positivity preserving?

In order to apply the diffusion maps in a matrix $C\in\mathbb R^{n\times n}$ , that matrix must obey some restrictions, C is symmetric: $C_{ij} = C_{ji}$, C is positivity preserving (PP): $\forall ...
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389 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? ...
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184 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 ...

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