# What is difference between $\hat{X}_n \overset{p}{\to} \bar{x}$ and $(\hat{X}_n - \bar{x}) = o_p(1)$?

Let $$\{\hat{X}_n\}$$ be a sequence of estimators that converges in probability to the constant $$\bar{x}$$, which I take to mean that, for any $$\epsilon > 0$$, $$\lim \limits_{n \to \infty} \Pr(|\hat{X}_n - \bar{x}| > \epsilon) = 0$$.

I have also seen people write $$(\hat{X}_n - \bar{x}) = o_p(1)$$ and have seen some people refer to it as equivalent to the statement that $$\hat{X}_n$$ is consistent for $$\bar{x}$$.

What is the difference between these two statements?

Is it correct that $$(\hat{X}_n - \bar{x}) = o_p(1)$$ is equivalent to $$\lim \limits_{n \to \infty} \Pr(|\hat{X}_n - \bar{x}| \geq \epsilon) = 0$$? And could one also write that $$(\hat{X}_n - \bar{x}) = o_p(1)$$ is equivalent to $$\hat{X}_n = \bar{x} + o_p(1)$$?

Thank you for any and all help!

Yes, $$X_n-\bar x=o_p(1)$$ is just notation for saying "For every $$\epsilon>0$$, $$\lim_{N\to\infty}P\left(|X_n-\bar x|>\epsilon \right)=0"$$

More generally, $$X_n-Z_n=o_p(Y_n)$$ is notation for saying "For every $$\epsilon>0$$, $$\lim_{N\to\infty}P\left(|X_n-Z_n|>\epsilon |Y_n| \right)=0"$$

And could one also write that $$(\hat{X}_n - \bar{x}) = o_p(1)$$ is equivalent to $$\hat{X}_n = \bar{x} + o_p(1)$$?

I do not know whether this is conventional, but you can do it.

### Interpretation in terms of quantile functions

We can argue about these notations with $$o_p$$ and $$O_p$$ in the same way as for $$o$$ and $$O$$ when we make an interpretation that connects the two.

With a particular interpretation you can turn the probabilistic expressions $${o}_p(f(n))$$ or $${O}_p(f(n))$$ into an expression with the more well known $${o}(f_n)$$ or $${O}(f_n)$$.

We can interpret

$$X_n = {o}_p(f(n)) \quad \text{for }n \to \infty$$

in terms of the quantile functions for $$X_n$$

$$\forall p: \lbrace Q_{X_n}(p) = {o}(f(n)) \quad \text{for }n \to \infty \rbrace$$

which means

$$\forall p: \lim_{n \to \infty} \frac{Q_{X_n}(p)}{f(n)} = 0$$

For example: in the figure below we show some quantiles functions of the mean $$\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$$, where $$X_i \sim N(\mu,\sigma^2)$$ with $$\mu = 0.5$$ and $$\sigma =1$$

The convergence of the quantile functions means that for every for any $$0 < p \leq 1$$, there is an $$n$$ such that $$Pr(\vert X_n \vert > \epsilon) \leq p$$

### Interpretation of equal sign

With this view, in terms of quantiles, the notations $$o_p$$ and $$O_p$$ follow the same rules as the notations $$o$$ and $$O$$ in 'regular' mathematics.

For these notations, there are some debates on how the equality sign '$$=$$' must/can be interpreted. See for instance

The equality sign '$$=$$' is not the typical equality and should be interpreted as a short-handed notation for 'is element of', or in terms of symbols '$$\in$$', and should not be interpreted as an 'identity'.

The acceptance and use of notations with an algebraic appearance like $$\hat{X}_n = \mu_{x} + o_p(1)$$ is a matter of taste, but it is certainly not wrong. You may however need to keep in mind that you do not confuse the reader.

To contemplate the meaning of $$\hat{X}_n = \mu_{x} + o_p(1)$$, the reader might need to imagine eventually the 'difference' $$\hat{X}_n - \mu_{x} = o_p(1)$$. So you could write it out like that from the start.

Which notation you use might depend on the intuition that you want to stress.

• The function is some other function plus/minus some bounds: $$f(x) = g(x) + O(h(x))$$
• The difference between the function and some other function is within some bounds: $$f(x) - g(x) = O(h(x))$$

To me it is a nuanced difference and the notations like $$f(x) = g(x) + O(h(x))$$ are not wrong. They are actually being used a lot (this sounds like an argumentum ad populum, but it is not a fallacy when we are arguing about conventions and use of interpretations that might be confusing when they are uncommon).