Abstract
Conditioned limit theorems as n →∞ are given for the increments X1, …, Xn of a random walk Sn = X1 +· · · + Xn, subject to the conditionings Sn ≥ nb or Sn = nb with b > EX. The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting (formula presented) of the increment density (formula presented). It has been noted in various formulations that conditionally, the increment density somehow is close to fθ (x). Sharp versions of such statements are given, including correction terms for segments (X1, …, Xk) with k fixed. Similar correction terms are given for the mean and variance of (formula presented) where (formula presented) is the empirical c.d.f. of X1, …, Xn. Also a result on the (total variation) distance for segments with k/n → c ∈ (0, 1) is derived. Further functional limit theorems for ̂(formula presented) are given, involving a bivariate conditioned Brownian limit.
| Original language | English |
|---|---|
| Journal | Bernoulli |
| Volume | 30 |
| Issue | 1 |
| Pages (from-to) | 371-387 |
| Number of pages | 17 |
| ISSN | 1350-7265 |
| DOIs | |
| Publication status | Published - Feb 2024 |
Keywords
- Boltzmann law
- conditioned Brownian motion
- empirical c.d.f
- exponential tilting
- functional limit theorem
- Gibbs conditioning
- saddlepoint approximation
- total variation distance