Question

Let $\text{Missing left or extra right}$ be a sequence of real numbers. Hence, we can also say that $\text{Missing left or extra right}$ is a sequence of constant (degenerate) random variables. Let $a$ be a real number. Show that $a_n\to a$ is equivalent to $a_n\stackrel{P}{\to }a$

Short answer

If $a_n$ is a sequence of constant random variables such that $a_n$ deterministically converges to 'a', it also stochastically converges to 'a'. This means that $a_n\to a$ is equivalent to $a_n\stackrel{P}{\to }a$.

Step-by-step solution

Define deterministic convergence

In the deterministic case, $a_n\to a$ means that as n goes to infinity, $a_n$ approaches 'a'. Formally, for every positive $ϵ$, there exists an integer N such that for every n larger than N, the absolute value of the difference between $a_n$ and 'a' is less than $ϵ$.

Define stochastic convergence

In the stochastic case, the convergence in probability of the sequence $a_n$ towards 'a', written as $a_n\stackrel{P}{\to }a$, means that the probability of this sequence deviating from 'a' by more than an arbitrarily-small $ϵ$ tends to zero as n tends to infinity. Formally, for every positive $ϵ$, the limit as n goes to infinity of the probability of the absolute value of the difference between $a_n$ and 'a' being greater than $ϵ$, is zero.

Show equivalence

We can see that the definitions are equivalent as long as we think of $a_n$ as a constant random variable that always equals the real number $a_n$. This means that the probability of $a_n$ deviating from 'a' by any positive number $ϵ$ is either 0 or 1: it's 0 if $a_n$ is within $ϵ$ of 'a', and 1 otherwise. Thus, when $a_n$ deterministically converges to 'a', it also stochastically converges to 'a', because the probability that $a_n$ deviates from 'a' by more than $ϵ$ (which is always 0 or 1) goes to zero, as required. Vice versa, if $a_n$ stochastically converges to 'a', it means that for every positive $ϵ$, $a_n$ will eventually be within $ϵ$ of 'a' with probability 1, which translates to $a_n$ deterministically converging to 'a'.