8
votes
On mathematical studies of the Mpemba effect
Since this question is still open, I take the liberty of pointing to a recent survey of the status of the Mpemba effect, Pathological Water Science -- Four Examples and What They Have in Common, which ...
- 188k
6
votes
Accepted
Is this a Brownian motion?
I vote for Mateusz Kwaśnicki. The condition for whether the random walk you generate this way scales towards Brownian motion under taking long times and rescaling is whether or not the variance is ...
- 471
5
votes
Accepted
Violating an order statistic inequality?
The DKW inequality can be stated as follows:
\begin{equation}
p_n(z):=P(\|S_n\|_\infty>z)\le 2e^{-2z^2} \tag{1}\label{1}
\end{equation}
for $z>0$, where
\begin{equation}
S_n(t):=\frac1{\...
- 128k
5
votes
Real world example of use of Monte Carlo method for high dimensional integrals
Some buzzwords that should lead to some non-textbook examples: In the fields of uncertainty quantification, statistical inverse problems or Bayesian inference one wants, for example, compute ...
- 12.7k
4
votes
Another question on provable non-existence of an efficient deterministic numerical method
There is a method, provided the times can be scaled to be integers not too large. Consider that the times are $t_1,\ldots,t_{40}$. The problem can be described like this: for some given $N$, how many ...
- 37.7k
4
votes
Intractability of an integral by deterministic numerical methods
For small $n$ Monte Carlo integration is not needed. For $n$ up to 100 see Kolmogorov-Smirnov Tests when Parameters are Estimated with Applications to Tests of Exponentiality and Tests on Spacings (...
- 188k
4
votes
Accepted
How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
The convergence of the discretized version $\max_{x \in \{0, \ldots, N\}} |W(x/N)|$ of $M:=\max_{0 \le t \le 1} |W(t)|$ to $M$ will be very slow -- at the rate of $1/\sqrt N$, according to Korolyuk ...
- 128k
4
votes
Accepted
How to draw a random normal matrix?
Any real normal matrix $M$ can be written as $M=O\,\mathrm{diag}(B_1,\ldots,B_\ell)\,O^t$ where $O$ is orthogonal and where the blocks $B_j$ are either $1\times 1$ real numbers or $2\times2$ matrices ...
- 2,135
4
votes
Accepted
What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
The extrema are attained for symmetric (isosceles) triangles. Trigonometry gives for the pairs (area, perimeter) as a function of the half tip angle $\psi$ the relations
$$\left( 4 \,\sin \psi\,\cos^3\...
- 1,676
3
votes
How to simulate the fractional noncentral Wishart distribution?
Efficient Simulation of the Wishart model (2009) shows how to simulate the fractional non-central Wishart distribution for all $\nu>p-1$. See section 6.1.2.b for $\nu\geq p+1$ and page 41 and ...
- 188k
3
votes
Another question on provable non-existence of an efficient deterministic numerical method
I tried Brendan's method on some data: $t_1 \ldots t_{40}$ are a sample of size $40$ from the exponential distribution with parameter $1$.
I want to bound the probability $p$ that $\sum_{i \in S} t_i ...
- 54.2k
2
votes
Real world example of use of Monte Carlo method for high dimensional integrals
Here are a few papers that discuss high-dimensional Monte Carlo integrals, together with quotes from Math Reviews.
MR2719643 (2011i:65038) Griebel, Michael; Holtz, Markus;
Dimension-wise ...
- 39.9k
2
votes
Design a Galton Board to simulate a uniform distribution
The method above to converge to the uniform distribution is in the right spirit to generalize the Galton board. A simple regular repeating pattern.
The following is much less aesthetic (and my crude ...
- 30.1k
2
votes
Accepted
Sampling uniformly in a ball of radius $\epsilon$ in the space of dicrete r.v. of m modalities for the total variation metric
For the uniform sampling from general convex polytopes, see e.g. Answer 1, Answer 2, other answers to this Question, and further references given there. Also, for instance, Fast MCMC Sampling ...
- 128k
1
vote
Simulation of multivariate logistic distribution conditional to a plane
According to your comment, your plane is $P=\{u\alpha+v\beta+b\colon(u,v)\in\mathbb R^2\}$, where $\alpha$ and $\beta$ are two orthogonal vectors in $\mathbb R^n$ and $b\in\mathbb R^n$. Assuming ...
- 128k
1
vote
Multiple Wiener-Ito integral distribution
Below some references regarding distributional properties of Wiener chaoses
The book, Gaussian Hilbert spaces, by S. Janson, is a standard reference to start with. In particular, you might want to ...
- 778
1
vote
Sampling from a particular multivariate probability distribution
Your question is a bit similar to a hanging post here How can we simulate from a geometric mixture?
To cite @whuber's comment under the SE post,
Without additional assumptions, this seems ...
- 8,071
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