Questions tagged [random-walk]

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

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Give a random walk on an interval with specified endpoints & extrema, can I find the probability that the max occurs before the min?

I have some summary measures on a time series process for a large number of time intervals, all of the same length. The summary measures are the initial value (i), which I will take to be zero without ...
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Is a random walk cointegrated with its own lag?

Can a random walk, or more broadly a unit-root process, be considered cointegrated with its own lag? E.g. if $y_t=y_{t-1}+u_t$ with $u_t\sim$ i.i.d., then $y_t$ is I(1), $x_t:=y_{t-1}$ is I(1) and ...
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Simple Symmetric Random Walk on $\mathbb{Z}$ is null recurrent

Question: Consider a simple symmetric random walk on integers, where from every state $i$ you move to states $i-1$ and $ i+1$ with probability half each. Show that this random walk is is null ...
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Random walk analysis of a univariate timeseries

As the title describes, I want to conduct a random walk analysis of a univariate time series $Y_t$. What are the tests and steps that you guys would suggest for this purpose? My current thinking: ...
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Random walks and martingales

In class, our professor explained that the martingale process is the in between case of random walk type I (innovations are i.i.d.) and random walk type II (innovations are serially uncorrelated). ...
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Different Random Walk Types and the Variance Ratio test

From my knowledge, we distinguish between two types of random walks. Type I, where the innovations are i.i.d. and Type II, where the innovations are serially uncorrelated. I want to conduct several ...
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Find the expected number of steps [duplicate]

Consider the following problem: Two people $M$ and $T$ walk over a straight line. The steps they take depend on flipping a coin: They move to the left if the result is head and they move to the right ...
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31 views

Random-walk and unit root processes predictable?

I know that a random walk is an AR(1) with a unit root, but there are also higher order autoregressive processes with unit roots. Does the unit root in such a higher order autoregressive process also ...
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Is every time series that is not predictable a random walk?

The title already reveals my question. I was wondering how specific the characterisitics of a random walk are defined and if every time series that is not predictable belongs to the class of random ...
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Weak form of market efficiency and random walks

The weak form of market efficiency states that historical prices should not provide predictive information that is not already incorporated in the current price. Hence, predictions based on the past ...
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Is ARIMA and Random Walk a Nested model?

I have confused about the Radom Walk. If a model restrict some parameter as 0 and the two regressions are the same, which is the Nested model. But in ARIMA model versus Random Walk model, the random ...
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19 views

Expected first passage time for random walk

A random walk on $\{0,1,2.....n\} $ with $p_{0,0} = p_{n,n} = 1$ and $p_{i,i+1} = p = 1-p_{i,i-1} = 1-q$ for $1 \leq i \leq n-1$ .Let $X_0 = i $ and $T$ be the first passage time to either 0 or $n$ ...
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What is the underlying proccess of the probability of a stock reaching a determined strike price?

Suppose the price of a stock follows a random walk, the price of a derivative "Stock is over $x at time t" would also follow a random walk? I talk about probabilities in the question ...
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Nyblom Harvey Test

I created in Matlab two independent random walk through the command: y = cumsum(randn(10000,2)) Although the series are clearly independent, when I try to run the ...
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Intuition of Random Walk having a constant mean

I am very new to time series analysis. A random walk is defined as $Y_t=\phi Y_{t-1}+\varepsilon_t$, where $\phi=1$ and $\varepsilon_t$ is white noise. It is said that process is non-stationary for ...
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SER/Confidence Interval of Random Walk Model?

Given the following random walk model $\Delta Y_t = Y_t - Y_{t-1}=\beta_0+u_t$, where $u_t \sim N(0,\sigma^2_u)$, how do we derive the standard error $\beta_0$ in terms of the estimated $\sigma^2_u$? ...
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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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1answer
40 views

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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33 views

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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1answer
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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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1answer
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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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63 views

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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1answer
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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
24 views

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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56 views

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
122 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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1answer
89 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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93 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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1answer
322 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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59 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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2answers
62 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 ...
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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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1answer
46 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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