If are pairwise independent and then as

Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

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# Tag: Probability

# Almost sure convergence via pairwise independence

# Expectation: Useful properties and inequalities

# Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

# Rio’s Inequality

# Kolmogorov’s Maximal Inequality

If are pairwise independent and then as

Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

Continue reading “Almost sure convergence via pairwise independence”

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If is a random variable on . The expected value of is defined as

**Inequalities**

*Jensen’s inequality*. If is convex and

*Holder’s inequality*. If with then

*Cauchy-Schwarz Inequality*: For

Continue reading “Expectation: Useful properties and inequalities”

**Definition 1.** A -field is a non-empty collection of subsets of the sample space closed under the formation of complements and countable unions (or equivalently of countable intesections – note ). Hence is a -field if

whenever

whenever

**Definition 2. Set functions and measures**. Let be a set and be an algebra on , and let be a non-negative set function

- is
**additive**if and, for , - The map is called
**countably additive**(or -additive) if and whenever is a sequence of disjoint sets in with union in , then - Let be a measurable space, so that is a -algebra on .
- A map is called a
**measure**on if is countable additive. The triple is called a**measure space**. -
The measure is called

**finite**ifand –

**finite**if, () s.th.

- Measure is called a
**probability measure**if and is then called a**probability triple**. - An element of is called -null if .
- A statement about points of is said to hold
**almost everywhere**(a.s.) if

** Continue reading “Very brief notes on measures: From σ-fields to Carathéodory’s Theorem” **

Let and be two integrable real-valued random variables and let be the quantile function of . Then if is integrable over we have

where is the a-mixing coefficient.

* Proof:* Set and then

since and

note also that

which implies that

- Let be independent random variables with . Set . Then
*Proof:*LetNotice that and