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 ...
Alexandre Eremenko's user avatar
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 ...
Terry Tao's user avatar
  • 108k
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 ...
Joseph O'Rourke's user avatar
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 ...
Iosif Pinelis's user avatar
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, ...
Joel David Hamkins's user avatar
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]:=\{...
Iosif Pinelis's user avatar
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 $c g_t$?
Bjørn Kjos-Hanssen's user avatar
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 ...
Yan Fyodorov's user avatar
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 ...
paul garrett's user avatar
  • 22.5k
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=...
Bjørn Kjos-Hanssen's user avatar
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(\...
Carlo Beenakker's user avatar
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 ...
Gerhard Paseman's user avatar
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 ...
Martin Hairer's user avatar
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_{...
Robert Israel's user avatar
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 ...
Liviu Nicolaescu's user avatar
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 ...
R W's user avatar
  • 16.6k
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 $c \in GF(p^n)^\ast,$ ...
kodlu's user avatar
  • 10.1k
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 ...
kodlu's user avatar
  • 10.1k
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 ...
Jean Duchon's user avatar
  • 3,055
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 $1/3$ ...
Bjørn Kjos-Hanssen's user avatar
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 ...
Carlo Beenakker's user avatar
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} ...
Liviu Nicolaescu's user avatar
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 ...
Carlo Beenakker's user avatar
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 ...
Fabrice Pautot's user avatar
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}[\...
Carlo Beenakker's user avatar
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 := \...
Dieter Kadelka's user avatar
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 ...
Iosif Pinelis's user avatar
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 $L(\alpha,G\cup M)$....
Mirco A. Mannucci's user avatar

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