# Questions tagged [random-functions]

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### 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
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### 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 ...
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### 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) ...
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### 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),...
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### Constructing k-wise independent variables over a general set

We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...
• 11
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### Does $E(XU)\neq 0$ imply $E(f(X) U)\neq 0$ "almost always"?

Consider two non-orthogonal random variables $$(1) \quad E(XU)\neq 0,$$ where $X$ can be a vector. Can we claim that (1) implies that $U$ will be "generically" non-orthogonal to any ...
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