14
votes
Accepted
Zeroes of a not quite holomorphic (but random if helpful) function
There is no general theory which applies here. However your problem can be restated
as a problem about zeros of harmonic maps, or about fixed points of anti-holomorphic maps, if you rewrite your ...
9
votes
Accepted
A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough.
Thanks for pointing out this ...
7
votes
Generating Random Curves with Fixed Length and Endpoint Distance
Here are some random "scribbles," based on
Bjørn Kjos-Hanssen's idea
(but not following his specifications exactly),
mixing with Izaak Meckler's comment:
What you see is points of a random walk fit ...
7
votes
Accepted
How to generate a random function with conditions?
You can simulate functions $u$ vanishing at $\infty$ using the formula
$$u(x)=u_N(x):=\sum_{n=0}^N \xi_n l_n(x),$$
where $N$ is a natural number, the $\xi_n$'s are independent standard normal (or ...
6
votes
Is it possible to check if a sequence of numbers (1 to 10) have been generated randomly or by a human?
Despite the comments, your question sits at the entryway to what has become an exciting new field of research.
In classical probability theory, if one were to flip a coin a certain number of times, ...
5
votes
Accepted
Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components
Consider the real-valued centered Gaussian process $(X_{t,a}\colon(t,a)\in T\times B_k)$, where
$$X_{t,a}:=\sum_{j\in[k]}a_j f_j(t),$$
$T:=\Omega$, $B_k$ is the unit ball in $\mathbb R^k$, and $[k]:=\{...
4
votes
Generating Random Curves with Fixed Length and Endpoint Distance
How about starting with a Brownian bridge $B_t$; then make it smooth and having finite length by convolution $g_t=B_t*f$; and then multiplying by a constant to get a unit length curve $c g_t$?
4
votes
Expected number of local minima of random polynomial in high dimensions
I believe some of these question about large n asymptotics were answered in the paper
Yan V. Fyodorov, Antonio Lerario, and Erik Lundberg
''On the number of connected components of random algebraic ...
4
votes
Is it possible to check if a sequence of numbers (1 to 10) have been generated randomly or by a human?
Recapping some aspects of @Joel David Hamkins' answer: "probability" by itself is not enough to explain "randomness", as it turns out. After all, all possible sequences of 100 heads-or-tails are ...
3
votes
Reference for Function-Valued Random Variables?
Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here.
For instance if $W_t$ is 1-dimensional standard Brownian motion at time $t$ and $$P(\forall x\,F(x)=x^2)=1$$ and $Y=...
3
votes
About concentration of eigenvalues values of a random symmetric matrix in a specific interval
A random PSD matrix $M$ can be constructed by taking $M=WW^T$, with the $n\times n$ matrix elements of $W$ i.i.d. with mean zero and variance $\sigma^2$. For $n\gg 1$ the marginal distribution $\rho(\...
3
votes
Can this criterion to indicate the randomness some numbers?
I find several problems with this.
The first problem is that you do not have a clear definition for the a(j) values. In particular, after reading your code, it seems a(2) depends not only on the ...
3
votes
Accepted
Attractors in random dynamics
Since this is a monotone continuous random dynamical system such that the underlying Markov process is uniquely ergodic, it admits a unique random fixed point. The standard proof goes as follows: by ...
3
votes
Accepted
Sign-oscillations for power series with random coefficients
We can construct inductively a sequence $t_n \to 1-$ and an increasing sequence $K_n$ of positive integers
such that with probability $> 1 - 1/n^2$, $f(t_n)$ has the same sign as
$V_n = \sum_{k=K_{...
3
votes
Random processes with smooth paths
A good place to read about this is Adler and Taylor's book Random Fields and Geometry. Regular random processes or functions are used more frequently in integral geometry.
Consider the more ...
3
votes
Accepted
Construct a random vector as a function of another random vector
I will begin with a reformulation of your question which makes it not only more symmetric, by also (at least for me) more natural and interesting. I will pass from your variables $(W,H,Q)$ to new ...
3
votes
Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$
Let $\mathrm{tr}:GF(p^n)\rightarrow GF(p)$ be the trace function which is equidistributed. Choose a basis so that you consider $GF(p^n)$ as the vector space $GF(p^n)$. For any $c \in GF(p^n)^\ast,$ ...
2
votes
Accepted
The best linear approximation of a random function
Firstly, I think you probably meant to take an absolute value in your maximum, otherwise by complementing the functions you'd get zero expectation, no?
F. Rodier has shown the following in the paper ...
2
votes
Generating Random Curves with Fixed Length and Endpoint Distance
Defining $z_t(s)=\int_0^s e^{itu(\sigma)}\ d\sigma$ transforms any function $u$ : $(0,1) \to \mathbb R$ into a one-parameter family of length $1$ curves $C_t=\{z_t(s):\ 0\leq s\leq1\}$ whose distance ...
2
votes
Accepted
Expected roots of polynomials with randomness in coefficients
Is $\mathbb{E}[\tilde{x}] = x$?
No --- in the case of just $x^3-d$, the root is $\sqrt[3]{d}$, and $E(\sqrt[3]{d})\ne\sqrt[3]{E(d)}$ typically. For instance let $d$ be 1, 8, 27 with probability $1/3$ ...
2
votes
Proving anti-concentration for the operator norm of a random matrix
Since the interest is in a PSD $X$, let me take $X=WW^{\rm T}$ with the elements of the $N\times M$ matrix $W$ i.i.d. with mean zero and variance $\sigma^2$. Note that the elements of $X$ itself are ...
2
votes
Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Here's a toy model that is truly linear
$$g(x) =\frac{1}{2}\sum_{i=1}^n \Lambda_i x_i^2, $$
where $\Lambda_i$ are i.i.d. $N(0,1)$ then
$$x(t)= \Big(e^{-t\Lambda_1} x_1(0),\dotsc, e^{-t\Lambda_n} ...
2
votes
Practical pseudorandom generators
The construction and validation of a PRG using one-way functions is the subject of
Practical Construction and Analysis of Pseudo-randomness Primitives
A different approach, using chaotic maps, is ...
2
votes
Practical pseudorandom generators
Cryptographically-secured PRNG's (CSPRNG) output bit sequences that are indistinguishable in polynomial time from "truly random sequences" (defined as having entropy rate 1bit/bit). They ...
2
votes
Partial derivative of expectation and Stein's lemma
Stein's lemma (with $Z_1=Z_2$) gives
$$\mathbb{E}[z\psi(g_1,\dots,g_{m-1},w+\tau z)] =\text{cov}\,(z,z)\mathbb{E}[\frac{d}{dz}\psi(g_1,\dots,g_{m-1},w+\tau z)]$$
$$\qquad=\tau\frac{d}{dw}\mathbb{E}[\...
2
votes
Positivity of linear combination of gaussian variables
Eventually not what you want. I have problems understanding your question. Let $b := \sum_{i=1}^n b_i > 0$ and $c^2 := \sum_{i=1}^n b_i^2$. Then $f \sim \mathcal{N}(b,p^2 \cdot c^2)$, thus $q := \...
1
vote
Emergence of non-power-law behaviour under infinite summing
First of all, it is easy to see that without loss of generality the dimension $n$ is $1$.
Next, of course $S:=\sum_i X_i$ can equal $\infty$ everywhere (if e.g. $X_i=1$ for all $i$), and then we will ...
1
vote
Forcing, constructibility, and random functions
Tim, here are my answers (low on technical but hopefully high on intuition):
The answer to the first question is YES, with one provision. You need to update the
$L(\alpha,G)$ with $L(\alpha,G\cup M)$....
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