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Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 9

Question

Math 6010 • Linear Models Fall 2016 • Question 9

Suppose \(X_1,N_1,X_2,N_2,\ldots \) are independent random variables such that the \(X\)’s are Bernoulli\((1/2)\)—that is, \(\mathbb {P}\{X_i=0\}=\mathbb {P}\{X_i=1\}=1/2\) for all \(i\geq 1\)—and each \(N_n\) is Poisson\((n)\); that is, \[ \mathbb {P}\{N_n=k\}=\frac {n^ke^{-n}}{k!} \qquad \text {for all integers }n\geq 1\text { and }k\geq 0. \] Find non-random sequences \(\{a_n\}_{n=1}^{\infty }\) and \(\{b_n\}_{n=1}^{\infty }\) such that, as \(n\to \infty \), \[ \frac {\sum _{i=1}^{N_n}X_i-a_n}{b_n}\Rightarrow \operatorname {N}(0,1). \]

Raw LaTeX

Suppose \(X_1,N_1,X_2,N_2,\ldots\) are independent random variables
such that the \(X\)'s are Bernoulli\((1/2)\)---that is,
\(\Prob\{X_i=0\}=\Prob\{X_i=1\}=1/2\) for all \(i\geq1\)---and each \(N_n\) is
Poisson\((n)\); that is,
\[
  \Prob\{N_n=k\}=\frac{n^ke^{-n}}{k!}
  \qquad\text{for all integers }n\geq1\text{ and }k\geq0.
\]
Find non-random sequences \(\{a_n\}_{n=1}^{\infty}\) and
\(\{b_n\}_{n=1}^{\infty}\) such that, as \(n\to\infty\),
\[
  \frac{\sum_{i=1}^{N_n}X_i-a_n}{b_n}\Rightarrow \operatorname{N}(0,1).
\]

Topic Tags

Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods