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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 2

Question

Math 6010 • Linear Models 1998 • Question 2

Let \(X_1,X_2,\ldots \) be independent identically distributed positive random variables with \(\E X_1=\mu \) and \(\operatorname {Var}(X_1)=\sigma ^2\). Let \[ N(t)=\min \left \{k:\sum _{1\le i\le k}X_i>t\right \},\qquad 0<t<\infty . \]

(a) Show that \[ \lim _{t\to \infty }\frac {1}{t}N(t)=\frac {1}{\mu } \] almost surely. (5 points)

(b) Show that \((N(t)-t/\mu )/(t\sigma ^2/\mu ^3)^{1/2}\) goes in distribution to a standard normal random variable. (5 points)

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent identically distributed positive
random variables with \(\E X_1=\mu\) and \(\var(X_1)=\sigma^2\). Let
\[
N(t)=\min\left\{k:\sum_{1\le i\le k}X_i>t\right\},\qquad 0<t<\infty.
\]

(a) Show that
\[
\lim_{t\to\infty}\frac{1}{t}N(t)=\frac{1}{\mu}
\]
almost surely. (5 points)

(b) Show that \((N(t)-t/\mu)/(t\sigma^2/\mu^3)^{1/2}\) goes in distribution to a
standard normal random variable. (5 points)

Topic Tags

Central Limit Theorems Convergence in Probability and Almost Sure Convergence Multivariate Normal Distributions Probability Inequalities and Tail Bounds