Questions tagged [random-walk]
A stochastic process that describes a path arising from a succession of random steps.
206
questions
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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 ...
2
votes
0answers
38 views
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 ...
1
vote
0answers
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 $R^2$ ...
0
votes
0answers
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 ...
4
votes
1answer
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 $[-1,1]$....
0
votes
0answers
5 views
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 ...
0
votes
0answers
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 ...
0
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0answers
6 views
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 ...
1
vote
2answers
45 views
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 ...
0
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0answers
18 views
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 ...
2
votes
1answer
40 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'...
1
vote
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 ...
0
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0answers
10 views
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 ...
0
votes
1answer
29 views
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$(y_t,y_{t+h} )=\sqrt{t ā (t+h)}$, for $t\ge 1$, $h>0$.
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11 views
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 Āµt+1 = Āµt + Ī²t + Ī·t, Ī²t+1 = Ī²t + Ī¶t, Ī¶t ā¼ NID(0,Ļ2 Ī¶), (3) where the ...
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0answers
18 views
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 ...
1
vote
0answers
8 views
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(-\...
0
votes
2answers
43 views
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 ...
0
votes
1answer
14 views
How to implement a survival function PDF?
I'm trying to generate a set of points whose separations are given by a LĆ©vy flight. A source that I have (Peebles 1993) says that the process goes as
Starting from a [point] in space, place the ...
3
votes
2answers
36 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,...
0
votes
1answer
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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0answers
28 views
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?
$$H_n=E[...
1
vote
0answers
20 views
Ornstein-Uhlenbeck process inside boundaries
I have some simulation of the Ornstein-Uhlenbeck process:
...
0
votes
1answer
47 views
How can I prove the simple random walk is a Markov process?
I know a simple random walk is defined as $X_t=X_{t-1}+w_t$, but how can I modify this equation is show it is a Markov process?
6
votes
2answers
1k views
Prove that a simple random walk is a martingale
Note that $a$ has a mean of 0.
My approach:
$$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 + ...
0
votes
0answers
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 ...
0
votes
1answer
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:
...
4
votes
1answer
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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0answers
25 views
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}...
1
vote
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-...
2
votes
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 ...
2
votes
2answers
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 ...
0
votes
0answers
25 views
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 $ā1\$$ with a 50% probability for either value, ...
10
votes
1answer
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 $ā1,...
1
vote
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.
0
votes
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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votes
0answers
14 views
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 ...
0
votes
1answer
36 views
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 ...
2
votes
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 ...
0
votes
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|...
1
vote
1answer
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}...
0
votes
0answers
41 views
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 ...
0
votes
0answers
30 views
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)...
0
votes
0answers
48 views
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 ...
3
votes
3answers
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 ...
0
votes
1answer
73 views
Proof for how the drift estimator, for a random walk with drift, is unbiased?
Random walk with drift formula is:
(Yt = Ī± + Yt-1 + Īµt )
How do I go about checking that the drift estimator Ī±-hat is unbiased.. which is proving that E(Ī±-hat) = Ī±?
Is this something I would need ...
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 ...
2
votes
0answers
21 views
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 ...
4
votes
2answers
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? ...
3
votes
2answers
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 ...
