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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 6

Question

Math 6010 • Linear Models Spring 2008 • Question 6

Suppose \(X_{1,n},X_{2,n},\ldots \) are i.i.d. with \(\mathbb {P}\{X_{1,n}=j\}=1/n\) for \(j=1,\ldots ,n\). Prove that if \[ T_n:=\inf \{k\geq 1:X_{1,n}+\cdots +X_{k,n}>n\}, \] then \(\lim _{n\to \infty }\mathbb {P}\{T_n=k\}\) exists for all \(k\geq 1\). Compute that limit.

Raw LaTeX

Suppose \(X_{1,n},X_{2,n},\ldots\) are i.i.d. with
\(\Prob\{X_{1,n}=j\}=1/n\) for \(j=1,\ldots,n\). Prove that if
\[
  T_n:=\inf\{k\geq1:X_{1,n}+\cdots+X_{k,n}>n\},
\]
then \(\lim_{n\to\infty}\Prob\{T_n=k\}\) exists for all \(k\geq1\). Compute that
limit.

Topic Tags

Laws of Large Numbers Uniform Integrability and Truncation Methods