Invariance principles for random walks conditioned to stay positive

By Francesco Caravenna, L. Chaumont. 2006

74 citations recorded by the source corpus.

Let {Sn} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (an) such that Sn/an con- verges in law to Y. Our main result is that the rescaled process (S⌊nt⌋/an, t ≥ 0), when conditioned to stay positive, converges in law (in the functional sense) to- wards the corresponding stable Levy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fe2f3731e4eb67af2ca0b2d9c9c02aa287f3b2c

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Citations in the corpus

Scaling limits for the peeling process on random maps (2014, background): “(ii) This follows from the invariance principle proved in [12].”
Conditional limit theorems for intermediately subcritical branching processes in random environment (2011, background): “From [13] (see also [12]) it follows that the distributions of L↓1 and L ∗ 1 are mutually absolutely continuous, therefore also the distribution ν of L1 has a density and is not concentrated on some compact interval.”
Branching random walks (2015, background): “Under the conditional probability Pa( · |Sλn ≥ 0), Sl σ l1/2 converges weakly (as l → ∞) to the Rayleigh distribution, whose density is ψ (see [83]).”
Local probabilities for random walks conditioned to stay positive (2007, background): “Itisnecessarytomentionthatfunctional limittheoremsforrandomwalksconditioned to stay positive were established by Doney [11] and by Caravenna and Chaumont [ 7 ].”
An invariance principle for random walk bridges conditioned to stay positive (2012, background): “More recently, such results have been extended to the case when the random walk is conditioned to stay positive for all time , cf. Bryn-Jones and Doney [10], Caravenna and Chaumont [13] and Chaumont and Doney [18].”