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”

Skip to content
# Tag: Statistics

# Almost sure convergence via pairwise independence

# An inequality of the mean involving truncation

# Some notes on Kalman Filtering

# Big Data for Volatility vs.Trend

# The limitations of randomised controlled trials

# Rio’s 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”

Advertisements

Let be i.i.d. r.vs with and . Then

Proof:

First we proove the following useful result

If and then

Note you can find the same lemma on Feller Vol.2 (p. 150) as

Continue reading “An inequality of the mean involving truncation”

**State Space form**

**Measurement Equation**

**Transition Equation**

**Future form**

So different aspects of Big Data — in this case dense vs. tall — are of different value for different things. Dense data promote accurate volatility estimation, and tall data promote accurate trend estimation.

In recent years, the use of randomised controlled trials has spread from labour market and welfare programme evaluation to other areas of economics (and to other social sciences), perhaps most prominently in development and health economics. This column argues that some of the popularity of such trials rests on misunderstandings about what they are capable of accomplishing, and cautions against simple extrapolations from trials to other contexts.

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