# Invariance principle: Brownian bridge and random walk conditioned on end point

Let $$\{X_i, i \in \mathbb{N}\}$$ be a sequence of non-lattice i.i.d. centered random variables, $$\mathbb{E} |X_1| ^3 < 0$$. Let $$S_n = \sum\limits _{i=1} ^n X_i$$ be the corresponding random walk and $$W^{(n)} _t = \frac{S_{\lfloor nt \rfloor}}{\sqrt{n}}$$, $$t \in [0,1]$$. I am looking for a reference that conditioned on the end point $$S_n$$, the normalized trajectory $$W^{(n)}$$ converges to a Brownian bridge: for a sequence $$a_n$$ such that $$\frac{a_n}{\sqrt{n}} \to a \in \mathbb{R}$$,

$$W^{(n)} \big| S_n \in [a_n, a_n+1] \Rightarrow B^a,$$ where $$(B_t, t\in [0,1])$$ is a Brownian bridge and $$B^a_t = B_t + at$$. In other words, for every continuous functional $$f : C[0,1] \to \mathbb{R}$$,

$$\mathbb{E} \Big \{ f (W^{(n)}) \Big| S_n \in [a_n, a_n+1] \Big\} \Rightarrow f(B^a).$$

A more general theorem is proved in  for the limits of random walks in the domain of attraction of a stable law. In the case described in the problem, one can also use the strong approximation approach in , because the probabilities of deviations of order $$\sqrt{n}$$ are smaller than the probabilities of the events that are being conditioned on.
• I have a very related question. If we condition $S_n=a_n$ in the sense of Doob, is there a reference for the Brownian bridge convergence? Let say $X_i$ are supported on the entire real line so that conditioning $S_n\in (a_n-\delta,a_n+\delta)$ always have a nontrivial probability. Nov 17, 2022 at 13:54