Notation almost sure convergence

Question

In an article they use a notation which I'm not familiar with and I appreciate your thoughts on this. It is about almost sure convergence. Suppose $C_i$ is a constant, $X_n$ and $Y_n$ are random variables with index $n$, and o denotes the usual Landau notation. For instance I have troubles with expressions like $X_n=C_1 +o(1)$ almost surely and $Y_n=C_2 +o(1)$ almost surely. I'm familiar with e.g. $X_n\to C_1$ almost surely for $n\to \infty$ or $X_n-C_1=o(1)$ almost surely. Their notation seems convenient because for expressions such as $X_n/Y_n$ they "plug-in" their above solution such that $X_n/Y_n=C_1/C_2+o(1)$ almost surely. I have never seen such a notation and appreciate your help. If any of you have seen similar notation let me know where. Thanks.

Answer 1

That "$X_n=C_1+o(1)$ almost surely" means that $X_n=C_1+Z_n$ where $P(Z_n\to0)=1$.

This is in line with the general principle that for every property $Q$, "$Q$ almost surely" means that $[Q\ \textrm{holds}]$ has full probability. Here, $$ [Q\ \textrm{holds}]=\{\omega\in\Omega\mid X_n(\omega)=C_1+o(1)\}=\{\omega\in\Omega\mid X_n(\omega)-C_1\to0\}. $$