Questions tagged [random-walk]

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

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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?
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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 + ...
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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 ...
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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: ...
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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 ...
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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}...
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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-...
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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 ...
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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 ...
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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, ...
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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,...
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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.
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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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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 ...
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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 ...
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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|...
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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}...
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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 ...
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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)...
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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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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 ...
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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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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: ...
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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 $1 −p$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 $x\in\mathbb R^d$. ...
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1answer
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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, & \...