# Questions tagged [random-walk]

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

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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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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$(y_t,y_{t+h} )=\sqrt{t ⁄ (t+h)}$, for $t\ge 1$, $h>0$.
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### 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 µt+1 = µt + βt + ηt, βt+1 = βt + ζt, ζt ∼ NID(0,σ2 ζ), (3) where the ...
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### 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 ...
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### 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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