Trjtdtk

Im a little confused about the difference of these two concepts, especially the convergence of probability. I understand that $X_n oversetpto Z $ if $Pr(|X_n - Z|>epsilon)=0$ for any $epsilon >0$ when $n rightarrow infty$.

I just need some clarification on what the subscript $n$ means and what $Z$ means. Is $n$ the sample size? is $Z$ a specific value, or another random variable? If it is another random variable, then wouldn't that mean that convergence in probability implies convergence in distribution? Also, Could you please give me some examples of things that are convergent in distribution but not in probability?

I will attempt to explain the distinction using the simplest example: the sample mean. Suppose we have an iid sample of random variables $X_i_i=1^n$. Then define the sample mean as $barX_n$. As the sample size grows, our value of the sample mean changes, hence the subscript $n$ to emphasize that our sample mean depends on the sample size.

Noting that $barX_n$ itself is a random variable, we can define a sequence of random variables, where elements of the sequence are indexed by different samples (sample size is growing), i.e. $barX_n_n=1^infty$. The weak law of large numbers (WLLN) tells us that so long as $E(X_1^2)<infty$, that
$$plimbarX_n = mu,$$
or equivalently
$$barX_n rightarrow_P mu,$$

where $mu=E(X_1)$. Formally, convergence in probability is defined as
$$forall epsilon>0, lim_n rightarrow infty P(|barX_n - mu| <epsilon)=1. $$
In other words, the probability of our estimate being within $epsilon$ from the true value tends to 1 as $n rightarrow infty$. Convergence in probability gives us confidence our estimators perform well with large samples.

Convergence in distribution tell us something very different and is primarily used for hypothesis testing. Under the same distributional assumptions described above, CLT gives us that
$$sqrtn(barX_n-mu) rightarrow_D N(0,E(X_1^2)).$$
Convergence in distribution means that the cdf of the left-hand size converges at all continuity points to the cdf of the right-hand side, i.e.
$$lim_n rightarrow infty F_n(x) = F(x),$$
where $F_n(x)$ is the cdf of $sqrtn(barX_n-mu)$ and $F(x)$ is the cdf for a $N(0,E(X_1^2))$ distribution. Knowing the limiting distribution allows us to test hypotheses about the sample mean (or whatever estimate we are generating).