A local limit theorem for random walks conditioned to stay positive

By Francesco Caravenna. 2004

71 citations recorded by the source corpus.

We consider a real random walk Sn=X1+...+Xn attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence Sn/an⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S1>0,...,Sn>0) and let Sn+ denote the random variable Sn conditioned on : it is known that Sn+/an ↠ ϕ+(x) dx, where ϕ+(x):=x exp (−x2/2)1(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X1 has an absolutely continuous law: in this case the uniform convergence of the density of Sn+/an towards ϕ+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c01214a80a2b172fa5cef9db5f48a3cd8f397ce8

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In Management Science and Operations Research

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Citations in the corpus

Invariance principles for random walks conditioned to stay positive (2006, background): “1 has been proven recently by BrynJones and Doney [7], by combining tightness arguments with a suitable local limit theorem for the conditioned random walk (that has been obtained independently also in [8]).”
Convergence in law of the maximum of nonlattice branching random walk (2014, methodology): “In particular, we do not use the convergence of the derivative martingale in the convergence in law proof, we do not use renewal theory (except to the extent that certain estimates from random walk, developed in [6], are used), and we do not work with the spine representation.”
Convergence in law of the minimum of a branching random walk (2011, unknown): “To deal with t ∈ [n1/2, (1 − λn)n], we see that 1{h≥−a}P ( min j∈[0,(1−λn)n−t] (−Sj ) ≥ 0, −S(1−λn)n−t ∈ [ (x − y) − a − h, (x − y)]) = 1{h≥−a}O(h + a + 1)((1 − λn)n − t + 1)−1 again by Theorem 1 of [12].”
An invariance principle for random walk bridges conditioned to stay positive (2012, background): “515 in [11], it follows that also for the constants in (3.”
Local probabilities for random walks conditioned to stay positive (2007, background): “For the case when the distribution of X belongs to the domain of attraction of the normal law, that is, when X ∈ D(2, 0) relation (5) has been proved by Caravenna [5].”
Branching random walks (2015, background): “By Theorem 1 of Caravenna [82] (this is where the non-lattice assumption is needed; the lattice case is also studied in [82], see Theorem 2 there), for l→ ∞,”
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].”
Convergence in law of the minimum of a branching random walk (2013, unknown): “[11] Caravenna, F.”