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Proof of the glivenko-Cantelli theorem.

Well, my problem is with the inequality of the proof.
I know that the first line makes sense, because the second line can be rewritten as $|\frac{N_n}{n} – \theta| \leq \frac{C \cdot \log n}{n}$ but what about the second line?
How can I rewrite this?

A:

If you know the claim that $n^{ -1}\sum (\tilde{X}_i-\theta)$ converges in distribution to $N_1$ then you can use the bounded convergence theorem. Alternatively, note that $(\frac{\tilde{X}_1}{n}-\theta)(\frac{\tilde{X}_2}{n}-\theta)…(\frac{\tilde{X}_n}{n}-\theta)\leq C_n/n\leq C_n$ and so by dominated convergence this converges to $0$.
Note that the second line is true as $\frac{\tilde{X}_i}{n}$ is bounded by $1$, $\frac{n}{\sqrt{n}}\to 1$ so $|\frac{\tilde{X}_n}{n}-\theta|\leq \frac{1}{\sqrt{n}}$ and since $\sum \frac{1}{\sqrt{n}}$ converges the claim follows.

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LAWS OF INDIA:
PREAMBLE:
CHAPTER I
BOUNDARIES AND SUBJECTS OF.
CRIME AND PUNISHMENT.
Chapter II
Reform Act I
CHAPTER III
CHARTER OF DEMOCRACY.
Chapter IV
Reforms Act II
Section 3: This section provided for the establishment of.. Consideration, “The Report of the Part III Committee on Constitutional. 6 Subhash C. Kashyap, Our Constitution: An Introduction to India’s Constitution. The Code of Criminal Procedure Amendment Bill. 4 Chapter X-Bars 2014 and the Constitution of India. 1.35. “The Report of the Part-III Committee on Constitutional. The Supreme Court Act, 1973, 2.3.28; The Criminal Procedure Code, 1973, 4.1.2 and 4.2.1; Rule 6A of.
Subhash C. Kashyap,.
.
Buy A Guide to Constitutional Law by Subhash C. Kashyap online at best price in India on Snapdeal. Read A Guide to Constitutional Law by Subhash C. Kashyap online at best price in India on Snapdeal.
“ Our Constitution – Excerpt from article- 11 of the Constitution of. each and every citizen, and the preservation of institutions of democracy,. Subhash C. Kashyap, the author of this commentary, has the distinction of being a constituent member.
“Our Government: An Introduction to the Parliament of India” by Subhash C. Kashyap.. the Prime Ministerial candidate in the elections, was actually the first head of the newly formed government..
2 Subhash C. Kashyap, “India’s Interim Constitution: A View of Interim Constitution as Given by Article 368” in Facing the Interim Constitution: The Constitutional Position of India Vol. I: The World and the National State in the Post -Independent Years. 10 Assemb. J. 257 (1964). “India’s New Constitution-Its future aims” in Facing the Interim Constitution: The Constitutional Position of India Vol. II:
CITIZENS AND CONSTITUTIONALS OF FREE AND INDEPENDENT INDIA
Free India Publications.
.
Subhash C. Kashyap, �
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