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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 10

Question

Math 6010 • Linear Models Fall 2013 • Question 10

Consider i.i.d. \(X_1,X_2,\ldots \) that are strictly positive, and define \[ S_n:=\sqrt {X_1+\sqrt {X_2+\sqrt {X_3+\sqrt {\cdots +\sqrt {X_n}}}}}, \qquad \text {for all }n\geq 1. \] In other words, if \(f(a,b):=\sqrt {a+\sqrt {b}}\) for all \(a,b>0\), then \(S_1:=f(X_1,0)\) and \(S_{k+1}:=f(S_k,X_{k+1})\) for all \(k\geq 0\).

Prove that \(\lim _{n\to \infty }S_n\) exists a.s. and is finite a.s.

Raw LaTeX

Consider i.i.d. \(X_1,X_2,\ldots\) that are strictly positive, and
define
\[
  S_n:=\sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\sqrt{\cdots+\sqrt{X_n}}}}},
  \qquad\text{for all }n\geq1.
\]
In other words, if \(f(a,b):=\sqrt{a+\sqrt{b}}\) for all \(a,b>0\), then
\(S_1:=f(X_1,0)\) and \(S_{k+1}:=f(S_k,X_{k+1})\) for all \(k\geq0\).

Prove that \(\lim_{n\to\infty}S_n\) exists a.s. and is finite a.s.

Topic Tags

Convergence in Probability and Almost Sure Convergence Laws of Large Numbers Uniform Integrability and Truncation Methods