# Questions tagged [random-walk]

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

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### Non anticipative sampling an ARIMA(1,1,0) process with known terminal value

I have an $\mathrm{ARIMA}(1,1,0)$ process $X_t$, for which I know the values $X_0=a$ and $X_T=b$. I want to sample paths $(X_t)_{t=1..(T-1)}$ consistent with the boundary conditions. One way to do it ...
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### When are continuous-time models important?

In Econometrics, majority of the models are in discrete-time setting, whereas when you move on to quantitative finance, continuous-time models are most prevalent. I get the theory and idea behind ...
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### Sampling with Random Walks

I feel so dumb having to post this, I pretty sure I'm just missing a "clever" rewrite of the problem. This isn't for homework, it's an old exam which I'm practising on; ...
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### What type of Markov Chain is a random walk of a Knight on a chessboard?

Assume we have the following chessboard and we have a knight that starts at the top left corner of the board. On every move the Knight chooses reachable square (i.e. a valid chess move a Knight can ...
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### Why can de-noising diffusion models be sampled with Gaussian distributions?

In de-noising diffusion models 1 the latent is typically sampled with a unit normal distribution, and then the sample (e.g. image) is generated by iteratively removing noise during the backwards ...
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### lowerbounding the expectation of maximum of $K$ random walking

Let $X_{i,j}$ be $K \times N$ i.i.d. random variables such that $P(X_{i,j} = 1) = P(X_{i,j} = -1) = \frac{1}{2}$, and $S_p = \sum_{q=1}^{N} X_{p,q}$ be $K$ i.i.d. random variables, each of which is an ...
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### Time-Series: Testing for random walks using the Hurst Exponent

Is the Hurst Exponent a good methodology for testing whether a series exhibits a random walk? I have read in some papers and websites that it is known for producing biased estimates and would like to ...
26 views

### Unbiased random walk : why is a random sample not calibrated

To simplify, consider unbiased random walks with absorbing barriers at 0 and 100. A random walk starting at X has an expected probability to hit the barrier 100 of exactly X%. However, it seems that ...
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1 vote
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### Time-Series: Testing for stationarity and random walks

My goal is to test the weak-form efficient market hypothesis using time-series on prices of various stocks listed on S&P 500. According to theory, a particular stock is said to be weak-form ...
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### The average distance in a 3D noncentral-chi-distribution random walk process after N steps

In the Noncentral Chi Distribution Wikipedia page, the calculated Mean is: $${\sqrt{{\pi\over2}}L_{1/2}^{(k/2-1)} \large( {\small{-\lambda^2\over2}})}$$ I am calculating the average distance after N ...
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### 3-D random walk: average distance after N steps

I am calculating the average distance in a 3-D random walk process after N steps. Each step is one unit long and the angle is randomly distributed around the origin. After N steps, what is the average ...
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### Time series: how much past predicts future

In financial (time series) statistics and forecasting we usually assume that the past of a series can predict the future to some extent. Every financial ad will warn you that investors should not ...
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### What can and can’t you say about a series with a unit root as evidenced by an ADF test? [duplicate]

I have a time series with 500+ observations which has a unit root, as evidenced by an ADF test at the sub 1% significance level. I want to explain to my class mates why that’s important and change the ...
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### Taking the Z transform and the Fourier transform of a Discrete-Time Random Walk

I saw that we could apply two transforms to the propagator of the Continuous-time random walk (CTRW), $$P(x,t)= \sum_{N=0}^\infty [\lambda^{*N}(x)w^{*N}(t)*\int_t^\infty d\tau w(\tau)]$$ where the ...
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### What is the difference between a Simple Random Walk and a Random Walk and why is one stationary, while the other is not?

To clarify, by a Simple Random Walk I mean $$Y_i = \begin{cases} -1 & prob = 1/2\\ 1 & prob = 1/2 \end{cases}$$ $$X_t = \sum_{i=1}^t{Y_i} \quad \textrm{,}\,X_0 = 0$$ and by Random Walk I ...
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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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1 vote
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### Generating function of a random walk

Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$. I would like to calculate the ...
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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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### 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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1 vote
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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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### 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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### 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 ...