36
votes
Accepted
Hamiltonian Monte Carlo vs. Sequential Monte Carlo
Hamiltonian Monte Carlo performs well with continuous target distributions with "weird" shapes. It requires the target distribution to be differentiable as it basically uses the slope of the target ...
- 494
32
votes
Intuition of Random Walk having a constant mean
To see what is happening you need more than one realisation of the random walk, because the mean and variance are summaries of the distribution of the walk, not of any single realisation.
This code ...
- 46.8k
30
votes
Why are random walks intercorrelated?
Your independent processes are not correlated! If $X_t$ and $Y_t$ are independent random walks:
A correlation coefficient unconditional on time does not exist. (Don't talk about $\operatorname{Corr}(...
- 23k
23
votes
Accepted
The magic money tree problem
This is a well-known problem. It is called a Kelly bet. The answer, by the way, is 1/3rd. It is equivalent to maximizing the log utility of wealth.
Kelly began with taking time to infinity and ...
- 7,810
19
votes
Explain what is meant by a deterministic and stochastic trend in relation to the following time series process?
The deterministic trend is one that you can determine from the equation directly, for example for the time series process $y_t = ct + \varepsilon$ has a deterministic trend with an expected value of $...
- 532
18
votes
Accepted
Why is a random walk not a stationary process?
For stationarity, the entire distribution of $p_t$ has to be constant over time, not only its mean. And while the mean of $p_t$ is indeed constant, e.g., it’s standard deviation isn’t. The larger $t$, ...
- 2,371
17
votes
Accepted
Intuition of Random Walk having a constant mean
There is a difference between unconditional mean and conditional mean, as there is between unconditional variance and conditional variance.
Mean
For a random walk
$$
Y_t=Y_{t-1}+\varepsilon_t
$$
with $...
- 69.5k
16
votes
Why are random walks intercorrelated?
The math needed to obtain an exact result is messy, but we can derive an exact value for the expected squared correlation coefficient relatively painlessly. It helps explain why a value near $1/2$ ...
- 334k
15
votes
Accepted
How to interpret ARIMA(0,1,0)?
ARIMA(0,1,0) is random walk.
It is a cumulative sum of an i.i.d. process which itself is known as ARIMA(0,0,0).
- 69.5k
12
votes
What is the Fourier Transform of a brownian motion?
Sorry, I know this thread is old, but I feel like some statements are not very clear and/or misleading, and also I would like to add a more mathematically sound perspective on the matter.
As was ...
- 357
12
votes
why does this tumbling tetrahedra result depend on n?
Jaynes says that if you were to know that the number of tosses so far was $n$ and that the length of the record was $y$ then you would know there had been $\frac{n-y}{2}$ annihilations.
The first ...
- 42.1k
10
votes
Accepted
Why is an unbiased random walk non-ergodic?
That Wikipedia article writes,
The process $X(t)$ is said to be mean-ergodic or mean-square ergodic in the first moment if the time average estimate $${\hat {\mu }}_{X}={\frac {1}{T}}\int _{0}^{T}X(...
- 334k
9
votes
Moving Average, Exponential Smoothing, and Random Walk for Forecasting
Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk model? (i.e. using only the most recent ...
- 69.5k
9
votes
The Dead Drunk Man
My question is, can the drunk man really "escape"? The man will always have a non-zero probability of returning to the starting point, albeit $0$.
Your random walk with unequal probability ...
- 86.5k
8
votes
How to interpret ARIMA(0,1,0)?
An ARIMA(0, 1, 0) series, when differenced once, becomes an ARMA(0, 0), which is random, uncorrelated, noise.
If $X_1, X_2, X_3, \ldots$ are the random variables in the series, this means that
$$X_{...
- 36.3k
8
votes
Accepted
Spurious Regressions (Random Walk)
Consider what random walks are: each new value is just a small perturbation of the old value.
When an explanatory variable $x_t$ and a synchronous response $y_t$ are both random walks, the pair of ...
- 334k
8
votes
Accepted
Prove that a simple random walk is a martingale
\begin{align}
E[X_{t+1} \mid X_1, \ldots, X_t]
&= E[X_t + a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t + E[a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t
\end{align}
- 2,338
7
votes
Accepted
Interpretation of an I(2) process?
One interpetation is that the rate of change is random walk.
It's like a free fall where the gravitational force is stochastically changing.
If you drop the body on earth, it's moving according to ...
- 62.3k
7
votes
What is the expected number of children until having the same number of girls and boys?
The answer by Matt F links to a sequence in the OEIS database, but provides no direct/clear motivation.
One can motivate the answer by considering a random walk with an absorbing boundary. You can ...
- 86.5k
7
votes
Interview Question: What is the probability they will be home in more than 30 minutes?
The student can only make it home in under 30 minutes by going straight there. The first coin flip effectively must come up heads, since the student will just stay at U re-flipping as many times as ...
- 10.4k
6
votes
How to implement a uniform random walk on a simplex?
I understand your simplex to be the specific subset
$$\Delta^d = \{(x_1,x_2,\ldots, x_d)\mid 0 \le x_1 \le x_2 \le \cdots \le x_d \le 1\} \subset \mathbb{R}^d.$$
(If not, a linear map onto any other ...
- 334k
6
votes
The magic money tree problem
I don't think this is much different from the Martingale. In your case, there are no doubling bets, but the winning payout is 3x.
I coded a "living replica" of your tree. I run 10 simulations. In ...
- 169
6
votes
What is the distribution of time's to ruin in the gambler's ruin problem (random walk)?
Time-to-ruin in the discrete-time random walk: From your question, I take it that you are referring to a discrete-time version of the gambler's ruin problem. Without loss of generality we can ...
- 133k
6
votes
why does this tumbling tetrahedra result depend on n?
The player sees the record after the last toss, not the value of the last toss. If the last toss lead to an annihilation then the last toss will not be in the record. Also, the player sees $y$, not ...
- 46.8k
5
votes
Accepted
Predictor for averaged Brownian motion
Let $B_t$ denote the state of the Brownian motion at each discrete time point $t=0,\pm 1,\pm 2,\dots$. Clearly, $B_t$ follows a random walk model
$$
(1-L)B_t = w_t \tag{1}
$$
where $L$ is the ...
- 11.7k
5
votes
How to prove that the probability of spurious correlation increases with random walk length?
This isn't a direct answer to your question but provides some pointers that relate to the asymptotic distribution of the Pearson correlation in a Gaussian random walk. (Neither replacing Spearman with ...
- 290k
5
votes
Accepted
Principal Components of Random Walk
I actually recently wrote a paper on this subject which will appear at NIPS 2018: https://arxiv.org/abs/1806.08805
My collaborator and I proved that in the limit of an infinite number of dimensions ...
- 2,164
5
votes
Accepted
Showing that R-squared might not be useful in time series data
Actually, for a random walk, we indeed have that $R^2\to_p1$, but not for the reason you posit.
Whether for a random walk or a stationary AR(1),
$$\frac {1}{T}\sum_{t=1}^T \hat u_i^2\to_p\sigma^2,$$
...
- 34.8k
Only top scored, non community-wiki answers of a minimum length are eligible