Local probabilities for random walks conditioned to stay positive

By V. Vatutin, Vitali Wachtel. 2007

125 citations recorded by the source corpus.

Let S0 = 0, {Sn, n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X1, X2, . . . and let $$\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$$ and $$\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $$. Assuming that the distribution of X1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as $${n\rightarrow \infty }$$, of the local probabilities $${\bf P}{(\tau ^{\pm }=n)}$$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities $${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$$ with fixed Δ and $${x=x(n)\in (0,\infty )}$$.

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

View Paper: https://www.semanticscholar.org/paper/ffaa8703d2cc6a70520c4538596f67562ca57d14

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Citations in the corpus

Random walks in cones (2011, background): “To prove a local limit theorem in the one-dimensional case, that is, for random walks conditioned to stay positive, one starts usually from the Wiener– Hopf factorization; see [8, 10, 45].”
On extrema of stable processes. (2010, background): “See also [25] for asymptotic results on distributions of functionals of a certain random walk related to stable processes and [24] for an explicit infinite series representation for the density of the first hitting time of a point in the spectrally positive case.”
Branching random walks (2015, background): “2 boil down to a special case of Lemma 20 of Vatutin and Wachtel [229].”
Local behaviour of first passage probabilities (2010, methodology): “The first statement is due to Erickson [18], and the second is a slight reformulation of Lemma 13 of [21], using the fact that nP (τ > n)P (τ > n) → k5, where τ − = min{n ≥ 1 : Sn ≤ 0} is the first weak decreasing ladder time.”
A simple path to asymptotics for the frontier of a branching Brownian motion (2011, methodology): “The estimates on Bessel processes used to estimate numbers of particles staying below straight lines can be replaced by small deviations probabilities for random walks conditioned to stay positive (see [20]); but calculating the expected number of particles staying below a curved line, our Lemma 10,”