Functional CLT for the Range of Stable Random Walks
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Functional CLT for the Range of Stable Random Walks Wojciech Cygan1,2 · Nikola Sandri´c3 · Stjepan Šebek4,5 Received: 21 January 2020 / Revised: 18 August 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this note, we establish a functional central limit theorem for the capacity of the range for a class of α-stable random walks on the integer lattice Zd with d > 5α/2. Using similar methods, we also prove an analogous result for the cardinality of the range when d > 3α/2. Keywords The range of a random walk · Capacity · Functional central limit theorem Mathematics Subject Classification 60F17 · 60F05 · 60G50 · 60G52
1 Introduction Let (, F, P) be a probability space, and let {ξi }i≥1 be a sequence of i.i.d. Zd -valued random variables defined on (, F, P), where Zd denotes the d-dimensional integer lattice. Further, let S0 = x, and Sn = Sn−1 + ξn , n ≥ 1, be a Zd -valued random walk starting from x ∈ Zd . The range of the random walk {Sn }n≥0 is defined as the random
Communicated by See Keong Lee.
B
Nikola Sandri´c [email protected] Wojciech Cygan [email protected] Stjepan Šebek [email protected]
1
Institut für Mathematische Stochastik, Technische Universität Dresden, Dresden, Germany
2
Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland
3
Department of Mathematics, University of Zagreb, Zagreb, Croatia
4
Institute of Discrete Mathematics, Graz University of Technology, Graz, Austria
5
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia
123
W. Cygan et al.
set Rn = {S0 , . . . , Sn }, n ≥ 0. Throughout the paper, we use the notation |Rn | to denote the cardinality of Rn . In addition to the cardinality of the range, we also consider its capacity. Let Px be the probability measure (on the space (, F)) which corresponds to {Sn }n≥0 starting at x ∈ Zd . We write P instead of P0 . For any A ⊆ Zd , we denote by T A+ the first hitting time of the set A by {Sn }n≥0 , that is, T A+ = inf{n ≥ 1 : Sn ∈ A}. + Also, when A = {x} for x ∈ Zd , we write Tx+ instead of T{x} . Recall {Sn }n≥0 is said to be transient if P(T0+ = ∞) > 0; otherwise, it is said to be recurrent, which implies that every random walk is either transient or recurrent. In the case when {Sn }n≥0 is transient, the capacity of a set A ⊆ Zd is defined as
Cap (A) =
Px (T A+ = ∞).
x∈A
For n ≥ 0, we denote Cn = Cap (Rn ). Observe that Cn is a random variable. The aim of this article is to prove a functional central limit theorem (FCLT) for the capacity and the cardinality of the range of the random walk {Sn }n≥0 , that is, for the stochastic processes {Cnt }t≥0 and {|Rnt |}t≥0 . The study on the range of random walks in Zd has a long history. A pioneering work is due to Dvoretzky and Erdös [8] where they obtained the strong law of large numbers for {|Rn |}n≥0 of a simple random walk in d ≥ 2. This result was later extended by Spitzer [23] to all random walks in d
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