Let (Yn)neN be i.i.d. random variables with Y₁ ~ N(0, 1). Denote by An the sample mean of th squares of these variables: An 1 == n n ΣY? i=1 eAn for all n E N. Define a sequence of random variables (Bn)neN by Bn := a) Does the sequence B₁ converge in probability to a constant b? If yes, what is b? b) Does the sequence Bn converge in distribution to a constant b? If yes, what is b?

MATLAB: An Introduction with Applications
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Let (Yn)neN be i.i.d. random variables with Y ~ N(0, 1). Denote by An the sample mean of the
squares of these variables:
1
An = -
n
i=1
ΣΥ
Define a sequence of random variables (Bn)nEN by Bn
:= eAn for all n E N.
a) Does the sequence Bn converge in probability to a constant b? If yes, what is 6?
b) Does the sequence Bn converge in distribution to a constant b? If yes, what is b?
Transcribed Image Text:Let (Yn)neN be i.i.d. random variables with Y ~ N(0, 1). Denote by An the sample mean of the squares of these variables: 1 An = - n i=1 ΣΥ Define a sequence of random variables (Bn)nEN by Bn := eAn for all n E N. a) Does the sequence Bn converge in probability to a constant b? If yes, what is 6? b) Does the sequence Bn converge in distribution to a constant b? If yes, what is b?
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