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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 5

Question

Math 6010 • Linear Models Spring 2020 • Question 5

Let \(\{X_n:n\in \mathbb {N}\}\) be a sequence of i.i.d. positive integrable random variables. Define \(T_n=X_1+\cdots +X_n\), \(n\in \mathbb {N}\), and \[ N(t)=\max \{n\in \mathbb {N}:T_n\leq t\}, \qquad t\geq 0. \] Show that with probability one, \[ \frac {N(t)}{t}\to \frac {1}{\mathbb {E}[X_1]} \qquad \text {as } t\to \infty . \] (Hint: Sketch what the process \(t\mapsto N(t)\) looks like for a fixed realization of the \(X_n\). What do you know about the asymptotic behavior of the jump times?)

Raw LaTeX

Let \(\{X_n:n\in\mathbb{N}\}\) be a sequence of i.i.d. positive
integrable random variables. Define \(T_n=X_1+\cdots+X_n\), \(n\in\mathbb{N}\),
and
\[
  N(t)=\max\{n\in\mathbb{N}:T_n\leq t\},
  \qquad t\geq0.
\]
Show that with probability one,
\[
  \frac{N(t)}{t}\to\frac{1}{\E[X_1]}
  \qquad\text{as } t\to\infty.
\]
(\textbf{Hint:} Sketch what the process \(t\mapsto N(t)\) looks like for a fixed
realization of the \(X_n\). What do you know about the asymptotic behavior of the
jump times?)

Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods