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Miklós Schweitzer 2012 P11

Source: Miklós Schweitzer 2012 P11

August 20, 2018

college contestsMiklos Schweitzerprobabilityreal analysis

Problem Statement

Let

X_1,X_2,..

be independent random variables with the same distribution, and let

S_n=X_1+X_2+...+X_n, n=1,2,...

. For what real numbers

c

is the following statement true:

P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}