$\frac{1}{\sigma \sqrt{n}} S_{\lfloor nt \rfloor}(\omega)$ is right continuous and has left hand limits

Question

Given random variables $\xi_1, \xi_2, \dots$ on a probability space, with partial sums $S_n = \xi_1 + \cdots + \xi_n$, let $X^n(\omega)$ be the function in the space of cadlag functions with value $$X_t^n(\omega) = \frac{1}{\sigma \sqrt{n}} S_{\lfloor nt \rfloor}(\omega).$$

How can we prove that this is a cadlag function for fixed $\omega$? Also is it right continuous and has left hand limits?

Answer 1

For any sequence $(a_n)$ of real numbers the function $f(t)=a_{[nt]}$ is cadlag. This is easy to verify from the definition of $[x]$.

Proof: Let $t_j$ decrease to $t$. Then $nt_j$ decrease to $nt$. If $k \leq nt <k+1$ then $k \leq nt_j < k+1$ for all $j$ sufficiently large so $[nt_j]=[nt]$ and $a_{[nt_j]} =a_{[nt]}$ for all $j$ sufficiently large. This proves right continuity. I leave the existence of left limits to you.