Questions tagged [random-functions]
The random-functions tag has no summary.
86 questions
2
votes
1
answer
130
views
Non-randomness of the image over finite fields of the truncated series development for the logarithm?
Given a prime number $p$ we consider the polynomial
$P_p(z)=\sum_{n=1}^{p-1} \frac{z^n}{n}$. For $p$ an odd prime $<1500$ the set
$P_p(\mathbb F_p))$ of values taken by $P_p$ on the finite field $\...
- 18.6k
11
votes
1
answer
403
views
Determine whether the random dynamical system $f(z)=1/(U-z)$ is bounded or not
I am looking at the random iteration in $\mathbb{C}$
$$z_{n+1}=\frac{1}{U_n-z_n},\qquad n\ge 0,$$
where $(U_n)$ is an random i.i.d. sequence taking the two complex values $4+\mathrm{i}$ and $4-\mathrm{...
- 311
3
votes
1
answer
133
views
Linear stability analysis of invariant distribution in a random dynamical system?
Consider the following discrete random dynamical system:
$$
X_{n+1} = F_\theta(X_n),
$$
where at each step the index $\theta \in \{1,2\}$ is chosen randomly and independently, and $X_n \in \mathbb{C}$....
- 311
0
votes
1
answer
171
views
Uniform concentration bound (function-valued random variable / continuous stochastic process)
I'm trying to consider a probability space $\Omega$ and
$f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...
- 321
0
votes
1
answer
261
views
Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$
A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...
- 121
3
votes
1
answer
312
views
How to generate a random function with conditions?
The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...
- 65
1
vote
1
answer
382
views
What is convergence in distribution of random variables taking values in a non-metrizable product space?
Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...
- 1,201
0
votes
1
answer
195
views
Lower bounding the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How ...
- 405
0
votes
2
answers
198
views
the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
- 405
2
votes
1
answer
246
views
Bounding random process
Def
$\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that
$$|X_t-X_s|\leq Cd(t,s),\text{ for all }t,s\in T.$$
Lemma
Suppose $\{X_t\}_{t\in T}$ is ...
- 405
2
votes
0
answers
137
views
Fourier expansion of random functions
Consider a random mapping $f:\{0,1\}^n \to \{0,1\}^n$, .i.e, a function such that for each $x \in \{0,1\}^n$, $f(x) \in \{0,1\}^n$ is chosen uniformly at random.
My question is what would the fourier ...
- 121
0
votes
1
answer
124
views
Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows
Let $T:=[-1,1]^{n-1}\times (0,1]$. Let
$$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$
where
(i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables
(ii) ...
- 149
0
votes
1
answer
164
views
Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
- 149
1
vote
1
answer
256
views
concentration of random field to its expectation function
Question
Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}_N$. Is there result regarding the concentration of the random field? For example
$\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\...
- 405
1
vote
0
answers
120
views
Expected Number of roots in $\mathbb D (0;r)$
In the literature about roots of random polynomials there are many results about the expected number of real roots of a complex polynomial building on Kac formula especially in the the asymptotic ...
- 838