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Normalisation

.ACS.

Consider the multiple regression model

\displaystyle y=X\beta +u

where

\displaystyle E(u)=0

and

\displaystyle V(u) = \sigma^2 I_n

where

\displaystyle \beta

is a

\displaystyle K \times 1

vector, and

\displaystyle X

is a full rank

\displaystyle n \times K

non-stochastic matrix.

By using the fact that

\displaystyle \lim_{n\to\infty}\sum_{i=1}^n i^\alpha = \lim_{n\to\infty}\int_1^n x^\alpha \, dx

for all

\displaystyle \alpha>0

Find a suitable normalisation for

\displaystyle \sum i

and for

\displaystyle \sum i^2

\displaystyle n \to \infty

and calculate the limit of the normalised sums.

I'll be truthful here... I've absolutely no idea what's being asked let alone how to proceed. Any help?

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Normalisation

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