Questions tagged [empirical-processes]
75 questions
9
votes
0 answers
$P(X_{(n-k_n)}>X_1\mid X_1>u_n)=0$?
Let $X_1,X_2,\dots$ be continuous random variables with full support (I need the result when they follow AR time series $X_i=\alpha X_{i-1}+\varepsilon_i$ for iid epsilons. But if you will consider iid case, it may also help with the time series…
Albert Paradek
- 897
- 1
- 7
- 19
7
votes
0 answers
Symmetrization argument for dependent variables
A standard argument in empirical process theory leads to the following inequality: let $Z_1, \dots, Z_n$ be i.i.d random variables and let $g$ be a convex function. Then it holds that
$$
\mathbb{E}\left[g\left( \sum_{i=1}^n Z_i - \mathbb{E}[Z_i]…
VHarisop
- 4,100
6
votes
1 answer
Does the law of large numbers hold for covering numbers?
I am self-studying empirical process theory.
I have encountered the covering number $N(\delta,\mathcal{G},P)$, as well as the empirical version $N(\delta,\mathcal{G},P_n)$.
It seems intuitive to expect some kind of…
Idontgetit
- 1,556
5
votes
0 answers
Proving a function of the empirical distribution is a Martingale
Let $X_1, \dots, X_n$ be a sequence of i.i.d. random variables with distribution function $G$, let
$$ G_t = \frac{\# \{ k : X_k \leq t \}}{n} $$
define the empirical distribution relative to the random variables. Set $A_t = \sqrt{n} (G_t -…
Jacob Denson
- 2,271
4
votes
0 answers
Empirical process symmetrization lower bound
This is part of Exercise 7.1.9 from Vershynin's book "High Dimensional Probability":
Suppose $X_1(t),\dots,X_N(t)$ are $N$ independent, mean zero random processes indexed by points $t\in T$. Let $\varepsilon_1,\dots,\varepsilon_N$ be independent…
George Giapitzakis
- 7,097
4
votes
2 answers
Union Bound of two events?
I am trying to understand the assumption proof of Theorem 2(Page -$7$) in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke.
Inequality 1
\begin{align}
\mathbb{P}\left(\frac{1}{n}…
user791678
4
votes
1 answer
Convergence of non iid observations on the empirical distribution
Let $f$ be a function on domain $X$ with binary output $f: X\to \{0,1\}$. We define an arbiatry distribution $\mathcal{Q}$ over $X$ and the empircal distribution of $n$ samples from $\mathcal{Q}$ -- $\mathbf{Q}^n$
By the Glivenko–Cantelli theorem…
user2757771
- 302
4
votes
1 answer
How to fit ordinary differential equations to empirical data?
For some biological systems, there exists ordinary or partial differential equations that allow one to simulate their activity/behavior over time. Some of these models even produce data that is very difficult to tell apart from real data.
What I…
mmh
- 243
- 2
- 7
3
votes
0 answers
A Donsker theorem in continuous time: weak convergence of $Z_T$ with $Z_T(f)=\frac{1}{\sqrt{T}}\int_0^Tf(X_t)dt$ over $\mathcal{F}\subset L^2$
Fix a filtered probability space $(\Omega,\mathscr{A}, (\mathscr{A}_t)_{t\geqslant 0},\mathbb{P})$ and let $(X_t)_{t\geqslant 0}:\Omega\times\mathbb{R}_+\to\mathbb{R}^d$ be an adapted stationary continuous time stochastic process (initialised at its…
Daan
- 646
3
votes
0 answers
Central limit theorem of independent partial sums
Let $M,N\to\infty$ with $M/N\to 0$. Suppose there is a random sequence $X_{mi}$ with mean 0 and variance $\sigma_{m}^{2}$. $X_{mi}$ is independent of $X_{mj}$ for all $i\neq j$ but $X_{mi}$ is correlated with $X_{m'j}$ for any $m\neq m'$ and all…
Ecthelion
- 165
- 6
3
votes
0 answers
Conditional Density Estimation in RKHS
I would like to model the conditional density of two real-valued random variable and estimate it using the empirical conditional mean embedding. I am not sure which of these two are correct way of doing this in RKHS.
A
If we model the joint density…
domath
- 1,254
3
votes
0 answers
Does the distribution of the maximum increase when adding independent Gaussian processes?
Let $x(t)$ and $y(t)$ be independent, mean-zero Gaussian processes, indexed over some general metric space $T$. Is it true that $\Pr(\sup_{t \in T} |x(t) + y(t)| > z) \ge \Pr(\sup_{t \in T} |x(t)| > z)$ for all $z > 0$?
Rob
- 441
3
votes
0 answers
Convergence of estimator defined by supremum over measurable sets
Let $X \in L_1$ be a positive random variable on the probability space $([0,1], \mathcal B, P)$, where $\mathcal B$ is the Borel $\sigma$ algebra on $[0,1]$. Consider
$$\phi(A) = E[X\mid A] \cdot I\big( P(A) \geq c \big),$$
for $A \in \mathcal B$…
northwiz
- 335
3
votes
2 answers
Discrete version of Dudley's inequalilty: Assuming set is finite wlog?
I'm self-studying Vershynin's High-dimensional probability book. I have a question about the proof of Theorem 8.1.4:
Let $(X_t)_{t\in T}$ be a mean-zero random process on a metric space $(T,d)$, with sub-gaussian increments.…
Idontgetit
- 1,556
