Finite-Dimensional Increments: Calculations and Proof Boundaries
The finite-dimensional increment identities and characteristic-function product calculations are written out. Vector-limit identification still invokes multivariate continuity and continuous-mapping tools, whose full development belongs to Chapter 14. This is not a path-space convergence proof.
F. Exact Finite-Dimensional Increment Identities and Joint Limits for Random Walks
The step process defined in Chapter 4 is W_n(t)=S_{\lfloor nt\rfloor}/\sqrt n, so
W_n(t_j)-W_n(t_{j-1})
=\frac1{\sqrt n}\sum_{k=\lfloor nt_{j-1}\rfloor+1}^{\lfloor nt_j\rfloor}\xi_kis an exact identity, not an approximation obtained by ignoring boundary terms. An error of at most two partial steps arises only when we switch to linear interpolation; for \pm1 steps, its absolute value is at most 2/\sqrt n.
Write m_{n,j}=\lfloor nt_j\rfloor-\lfloor nt_{j-1}\rfloor; then m_{n,j}/n\to t_j-t_{j-1}>0. Independence and the preceding characteristic-function expansion give, for any u_1,\ldots,u_m,
\mathbb E\exp\left(i\sum_{j=1}^mu_j\Delta_{n,j}\right)
=\prod_{j=1}^m\left[\varphi_\xi\left(\frac{u_j}{\sqrt n}\right)\right]^{m_{n,j}}
\longrightarrow\exp\left(-\frac12\sum_{j=1}^mu_j^2(t_j-t_{j-1})\right).This proves convergence of the joint distribution of the increment vector. Applying the continuous linear transformation that “adds up the increments” then gives finite-dimensional convergence of the vector of process values. This is still not weak convergence on the entire path space; that also requires tightness.