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Concentration

In real life, for the most part, we can’t compute probabilities in closed form. Instead, we either bound them, or we want to show that
P(A)0P(A) \approx 0
or
P(A)1P(A) \approx 1
.

Concentration Inequalities

Theorem 17 (Markov's Inequality)

For a non-negative random variable
XX
,
Pr{Xt}E[X]t,t0.\text{Pr}\left\{X \geq t\right\} \leq \frac{\mathbb{E}\left[X\right] }{t}, \quad t \geq 0.

Theorem 18 (Chebyshev's Inequality)

If
XX
is a random variable, then
Pr{XE[X]t}Var(X)t2.\text{Pr}\left\{|X - \mathbb{E}\left[X\right] | \geq t\right\} \leq \frac{\text{Var}\left(X\right) }{t^2}.
Intuitively, Theorem 18 gives gives a “better” bound than Theorem 17 because it incorporates the variance of the random variable. Using this idea, we can define an even better bound that incorporates information from all moments of the random variable.

Definition 36 (Chernoff Bound)

For a random variable
XX
and
aRa\in\mathbb{R}
,
Pr{Xa}E[etX]eta=etaMx(t).\text{Pr}\left\{X \geq a\right\} \leq \frac{\mathbb{E}\left[e^{tX}\right] }{e^{ta}} = e^{-ta}M_x(t).
After computing the Chernoff bound for a general
tt
, we can then optimize over it to compute the best bound possible.

Convergence

The idea of convergence brings the mathematical language of limits into probability. The fundamental question we want to answer is given random variables
X1,X2,X_1, X_2, \cdots
, what does it mean to compute
limnXn.\lim_{n\to\infty}X_n.
This question is not as straightforward as it seems because random variables are functions, and there are many ways to define the convergence of functions.

Definition 37

A sequence of random variables converges almost surely to
XX
if
P(limnXn=X)=1P\left(\lim_{n\to \infty}X_n = X\right) = 1
One result of almost sure convergence deals with deviations around the mean of many samples.

Theorem 19 (Strong Law of Large Numbers)

If
X1,X2,,XnX_1, X_2, \cdots, X_n
are independently and identically distributed to
XX
where
E[X]<\mathbb{E}\left[X\right] < \infty
, then
1niXi\frac{1}{n}\sum_i X_i
converges almost surely to
E[X]\mathbb{E}\left[X\right]
.
The strong law tells us that for any observed realization, there is a point after which there are no deviations from the mean.

Definition 38

A sequence of random variables converges in probability if
ϵ>0,limnP(XnX>ϵ)=0\forall \epsilon > 0, \quad \lim_{n\to\infty}P(|X_n - X| > \epsilon) = 0
Convergence in probability can help us formalize the intuition that we have which says probability is the frequency with which an even happens over many trials of an event.

Theorem 20 (Weak Law of Large Numbers)

Let
X1,X2,,XnX_1, X_2, \cdots, X_n
be independently and identically distributed according to
XX
, and let
Mn=1nXiM_n = \frac{1}{n}\sum X_i
. Then for
ϵ>0\epsilon > 0
,
limnPr{MnE[X]>ϵ}=0.\lim_{n\to\infty} \text{Pr}\left\{|M_n - \mathbb{E}\left[X\right] | > \epsilon\right\} = 0.
It tells us that the probability of a deviation of
ϵ\epsilon
from the true mean will go to 0 in the limit, but we can still observe these deviations. Nevertheless, the weak law helps us formalize our intuition about probability. If
X1,X2,,XnX_1, X_2, \cdots, X_n
are independently and identically distributed according to
XX
, then we can define the empirical frequency
Fn=1XiBn    E[Fn]=P(XB).F_n = \frac{\sum\mathbb{1}_{X_i\in B}}{n} \implies \mathbb{E}\left[F_n\right] = P(X \in B).
By Theorem 20,
limnPr{FnP(XB)>ϵ}=0,\lim_{n\to\infty}\text{Pr}\left\{|F_n - P(X\in B)| > \epsilon\right\} = 0,
meaning over many trials, the empirical frequency is equal to the probility of the event, matching intuition.

Definition 39

A sequence of random variables converges in distribution if
limnFXn(x)=Fx(x).\lim_{n\to\infty}F_{X_n}(x) = F_x(x).
An example of convergence in distribution is the central limit theorem.

Theorem 21 (Central Limit Theorem)

If
X1,X2,X_1, X_2, \cdots
are independently and identically distributed according to
XX
with
Var(X)=σ2\text{Var}\left(X\right) = \sigma^2
and
E[X]=μ\mathbb{E}\left[X\right] = \mu
, then
limnP(i=1nXinμσnx)=Φ(x)\lim_{n\to\infty}P\left(\frac{\sum_{i=1}^nX_i - n\mu}{\sigma\sqrt{n}} \leq x\right) = \Phi(x)
In other words, a sequence of random variables converges in distribution to a normal distribution with variance
σ2\sigma^2
and mean
μ\mu
.
These notions of convergence are not identical, and they do not necessarily imply each other. It is true that almost sure convergence implies convergence in probability, and convergence in probability implies convergence in distribution, but the implication is only one way.
Once we know how a random variable converges, we can then also find how functions of that random variable converge.

Theorem 22 (Continuous Mapping Theorem)

If
ff
is a continuous function, then if
XnX_n
converges to
XX
, then
f(Xn)f(X_n)
converges to
f(X)f(X)
. The convergence can be almost surely, in probability, or in distribution.