Multidimensional Chebyshev’s inequality
Let X be an N-dimensional random variable with mean μ=𝔼[X] and covariance matrix
V=𝔼[(X-μ)(X-μ)T].
If V is invertible (i.e., strictly positive), for any t>0:
Pr(√(X-μ)TV-1(X-μ)>t)≤Nt2 |
Proof: V is positive, so V-1 is. Define the random variable
y=(X-μ)TV-1(X-μ) |
y is positive, then Markov’s inequality holds:
Pr(√(X-μ)TV-1(X-μ)>t)=Pr(√y>t)=Pr(y>t2)≤𝔼[y]t2 |
Since V is symmetric, a rotation
R (i.e., RRT=RTR=I) and a diagonal matrix
D (i.e., i≠j⇒Di,j=0) exist such that
V=RTDR |
Since V is positive Dii>0. Besides
V-1=R-1D-1(RT)-1=RTD-1R |
clearly [D-1]ii=1Dii.
Define Z=R(X-μ).
The following identities hold:
𝔼[ZZT]=R𝔼[(X-μ)(X-μ)T]RT=RRTDRRT=D |
and
then
Title | Multidimensional Chebyshev’s inequality |
---|---|
Canonical name | MultidimensionalChebyshevsInequality |
Date of creation | 2013-03-22 18:17:55 |
Last modified on | 2013-03-22 18:17:55 |
Owner | daniWk (21206) |
Last modified by | daniWk (21206) |
Numerical id | 5 |
Author | daniWk (21206) |
Entry type | Theorem |
Classification | msc 60A99 |