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 $f : C[0,1] \to \mathbb{R}$,

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


1 Answer 1


A more general theorem is proved in [1] 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 [2], because the probabilities of deviations of order $\sqrt{n}$ are smaller than the probabilities of the events that are being conditioned on.

[1] Liggett, Thomas M. "An invariance principle for conditioned sums of independent random variables." Journal of Mathematics and Mechanics 18, no. 6 (1968): 559-570. https://www.jstor.org/stable/pdf/24901780.pdf

[2] Csörgo, Miklos, and Pál Révész. Strong approximations in probability and statistics. Academic press, 2014.

  • $\begingroup$ 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. $\endgroup$
    – Sayan
    Nov 17, 2022 at 13:54

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