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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 7

Question

Math 6010 • Linear Models 2004 • Question 7

Let \(\{X_n\}_{n=1}^{\infty }\) be a sequence of i.i.d. random variables with \(\mathbb {E}[X_1]=0\) and \(\mathbb {E}[X_1^2]=1\). Define \(S_k=X_1+\cdots +X_k\) for all \(k\ge 1\). Let \(N_m\) be another independent random variable with a mean-\(m\) Poisson distribution. Prove that there exist non-random \(\alpha _m\) and \(\beta _m\) such that \((S_{N_m}-\alpha _m)/\beta _m\) converges weakly to a standard-normal distribution as \(m\to \infty \). Describe an explicit example of \(\alpha _m\) and \(\beta _m\).

Raw LaTeX

Let \(\{X_n\}_{n=1}^{\infty}\) be a sequence of i.i.d. random variables
with \(\E[X_1]=0\) and \(\E[X_1^2]=1\). Define \(S_k=X_1+\cdots+X_k\) for all \(k\ge 1\).
Let \(N_m\) be another independent random variable with a mean-\(m\) Poisson
distribution. Prove that there exist non-random \(\alpha_m\) and \(\beta_m\) such that
\((S_{N_m}-\alpha_m)/\beta_m\) converges weakly to a standard-normal distribution as
\(m\to\infty\). Describe an explicit example of \(\alpha_m\) and \(\beta_m\).

Topic Tags

Covariance and Independence Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits