KarlRosaen


  1. ML Study
  2. Homework
  3. all-of-statistics-ch5-big-o-proofs

Problem

Prove or disprove the following:

(i) If $X_n = O_P(a_n)$ and $Y_n = O_P(b_n)$ then $X_n Y_n = O_P(a_n b_n)$

(ii) If $X_n = O_P(a_n)$ and $Y_n = O_P(b_n)$ then $X_n + Y_n = O_P(max(a_n, b_n))$

(iii) If $X_n = o_P(a_n)$ and $Y_n = o_P(b_n)$ then $X_n Y_n = o_P(a_n)$

(iv) If $X_n = o_P(a_n)$ and $Y_n = o_P(b_n)$ then $X_n + Y_n = O_P(a_n + b_n)$

(v) If $X_n = o_P(a_n)$ and $Y_n = O_P(b_n)$ then $X_n / Y_n = O_P(a_n / b_n)$

(vi) If $X_n = O_P(a_n)$ and $Y_n = o_P(b_n)$ then $X_n Y_n = o_P(a_n b_n)$

Note: the notation $O_p$ and $o_p$ is summarized on wikipedia as well as within 36-705 lecture 2 notes.

Solution

Sources