Questions tagged [random-walk]
A stochastic process that describes a path arising from a succession of random steps.
190
questions
0
votes
1answer
37 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
989 views
Prove that a simple random walk is a martingale
Note that $a$ has a mean of 0.
My approach:
$$X_t=X_{t-1}+a_t$$
$$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$
$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + ...
0
votes
0answers
30 views
How to make random walk stationary?
I know that random walk is modeled as; $Y_t = Y_{t-1}+ u_t$ ;
It is known that it is not a stationary model since the variance of observations at different times are not same. As a general, remedy ...
0
votes
1answer
60 views
Random walk based on coin toss [closed]
You are in an open field with two coins and you alternate flipping the coins and taking steps by the following rules:
...
2
votes
1answer
52 views
Random Walk Metropolis Hastings implementation in R using log scale
Context
I looked literally everywhere but I couldn't find a complete implementation of the Random Walk Metropolis-Hastings algorithm using the log scale. By log scale I mean that we are working with ...
0
votes
0answers
15 views
Use of the inversion method in sequential sampling to “invert” a random walk
Let $M\subseteq\mathbb R^3$ be Borel measurable, $\lambda$ be a $\sigma$-finite measure on $\mathcal B(M)$, $k\in\mathbb N$, $I:=\{0,\ldots,k\}$, $q$ be a probability density on $\left(E^I,{\mathcal E}...
1
vote
1answer
51 views
Mean and variance of a Random Walk with lower boundary
Consider a random walk $S_n$ that starts at $S_0$ and terminates if it hits a lower boundary of $0$.
$$
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
32 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
47 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
16 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 $−1\$$ with a 50% probability for either value, ...
10
votes
1answer
217 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,...
1
vote
1answer
39 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
40 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 ...
0
votes
0answers
13 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
32 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
53 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
26 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
87 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
34 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
27 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
44 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
131 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
57 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 ...
0
votes
0answers
6 views
How to generate a log-linear frequency distribution of walk durations with a random walk?
Imagine that I have a random walk of n-individuals running to time t. There is a lower absorbing boundary z at which each individual stops walking when encountered. Steps are randomly drawn from any ...
0
votes
0answers
7 views
Testing for weak-form efficiency - are unit root tests redundant after autocorrelation test?
I am testing the variability of the market efficiency over the time and one of the methods I'm using is a quite simple estimation of correlation coefficients and their significance for different lags (...
4
votes
0answers
166 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 ...
0
votes
0answers
23 views
Troubles with Basket-Sensitive Recommendation System using CF + Modified Random Walk
I'm trying to reproduce and understand the "Basket-Sensitive Random Walk" for recommendation systems proposed in the paper "Grocery Shopping Recommendations Based on Basket-Sensitive Random Walk" of ...
2
votes
0answers
15 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 ...
3
votes
2answers
173 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
101 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 ...
0
votes
1answer
21 views
How would a drift diffusion model explain effects in reaction time but not accuracy?
How would a drift-diffusion model explain a case where a variable has an effect in reaction time but not in accuracy. I know that one explanation would be a speed-accuracy tradeoff, in which the ...
0
votes
0answers
68 views
Dickey-Fuller test interpretation (urca package)
I am having trouble with interpreting the Dickey-Fuller test on a time series using the ur.df() function in the urca package. I already read this thread but still need some advise.
The command is:
...
1
vote
1answer
83 views
random walk on Z towards the origin
Consider a random walk on $\mathbb{Z}$ with rate $a>0$ (begin no origin). The r.w. jumps one step towards the origin with probability $p$ or one step away from the origin with probability $1 −p$. ...
0
votes
0answers
229 views
Calculating Conditional Expected Value in R
I have a stock whose returns follows a Random Walk with
mu= 6% and sigma= 20%
I would like to calculate the 10 returns of this distribution that I would get ...
1
vote
0answers
150 views
Conditional probabilities for sequence of events
The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes ...
2
votes
1answer
32 views
How to calculated a probability for a sum of integers to be equal to a given value if probabilities of addends are known?
Let's assume that we have a process generating some integer numbers with different probabilities. The set of possible integers is small. For example: -2, -1, 0, 1 and 2. We also know probabilities for ...
0
votes
1answer
90 views
Are all non-stationary series random walks?
Are all (non-explosive) time series either stationary around a deterministic trend or random walks?
If I run the ADF test and I can't reject the null of non-stationarity does it imply the series is ...
1
vote
0answers
51 views
Can the first difference of a time series follow a random walk?
I have a time series in levels, say $X_t$ and a variance-ratio test suggests that it does not follow a random walk.
Now I differenced the time series once and the variance-ratio test suggests that the ...
0
votes
0answers
8 views
Popular stochastic model for behavior of staying at the same position?
I am looking for a popular stochastic model employed for a trajectory of a fish which tries to keep staying at the initial position against water pressure from time-varying directions.
The trivial ...
0
votes
1answer
34 views
Confusing in Random Walk
I have a question about of random walk.
Consider a particle starting its random walk at 0. At each step, it either moves in positive or negative direction. If the probability of moving in positive ...
20
votes
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 ...
0
votes
0answers
65 views
Integrated Random Walk is invertible and/or stationary?
I am wondering if the integrated Random Walk $y_t$ is Stationary and/or invertible $ y_t = 2y_{t-1} - y_{t-2} + e_t $ where $ e_t$ is a White Noise.
To prove non stationarity I tried to say: If, by ...
6
votes
1answer
110 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 $x\in\mathbb R^d$.
...
1
vote
1answer
99 views
Average time for a random walk on the edges of a cube
In a interview i had something similar to this question Random walk on the edges of a cube.
But this time there is only a ant, it takes one minute to go through a single edge. The ant can use the ...
0
votes
0answers
17 views
Random walks-heavy tailed case
Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution
$P(\xi_{1}=-1)=1-C_{\beta}$ and $...
2
votes
1answer
32 views
User-Specific Activity Level Along Weeks
I am assigning Low | Medium | High activity levels to users once a week.
At the end of an entire period,a user has been assign n weekly activity levels among {Low, Medium, High}.
Let k be the total ...
2
votes
1answer
110 views
Probability you end up at the origin after taking $2n$ steps?
Starting at the origin on the line we take a step of unit to the left or to the right
with probability $\frac12$. We do this repeatedly with independent steps. If we take $2n$ steps, what is the
...
1
vote
0answers
140 views
Expected number of steps in Gambler's ruin game with two players
Let's say we have two players A and B. Player A has 3 coins and player B has 5 coins. If player wins the other player gives one coin. During game second player probability of loosing is $2/3$, while ...
5
votes
0answers
119 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 ...
1
vote
1answer
72 views
Why is the probability of a random walk reaching 1 (in n steps) squared greater than the probability of it reaching 2 (in n steps)?
Let $S_n$ be a simple random walk. i.e.
$$
S_n = \sum_{t=1}^n X_t,
$$
where ${X_t}$ are i.i.d random variables with
$$
X_t =
\begin{cases}
+1, & \textrm{w/ probability } p \\
-1, & \...
