Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$ Z d
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Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in Zd Inna M. Asymont1 · Dmitry Korshunov2 Received: 16 May 2019 / Revised: 14 August 2019 © The Author(s) 2019
Abstract For an arbitrary transient random walk(Sn )n≥0 in Zd , d ≥ 1, we prove a strong law of large numbersfor the spatial sum x∈Zd f (l(n, x)) of a function f of the local n I{Si = x}. Particular cases are the number of times l(n, x) = i=0 (a) visited sites [first considered by Dvoretzky and Erd˝os (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function f (i) = I{i ≥ 1}; (b) α-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to f (i) = i α ; (c) sites visited by the random walk exactly j times [considered by Erd˝os and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where f (i) = I{i = j}. Keywords Transient random walk in Zd · Local times · Strong law of large numbers Mathematics Subject Classification (2010) 60G50 · 60J55 · 60F15
1 Introduction and Main Results Let X 1 , X 2 , …be a sequence of independent identically distributed random vectors valued in Zd , d ≥ 1. Consider a random walk generated by X n ’s, S0 := 0, Sn := X 1 + · · · + X n , and the number of visits to a site x ∈ Zd up to time n which is called the local time of x,
B
Dmitry Korshunov [email protected] Inna M. Asymont [email protected]
1
Financial University under the Government of the Russian Federation, Moscow, Russia
2
Lancaster University, Lancaster, UK
123
Journal of Theoretical Probability
l(n, x) :=
n
I{Si = x}.
(1)
l α (n, x), α ≥ 0.
(2)
i=0
Define random variables
L n (α) :=
x∈Zd :
l(n,x)>0
In particular, the L n (0) = |{S0 , . . . , Sn }| represents the number of distinct sites visited by the random walk up to time n, called the range of (Sn )n≥1 . The case α = 1 is trivial because L n (1) = n + 1. The value of L n (2) is the number of so-called selfintersections of a random walk. For an integer α, the value of L n (α) is the number of α-fold self-intersections up to time n. It is known that for a recurrent random walk the quotient L n (0)/n tends to be 0 as n → ∞ (see, e.g. Spitzer [12, Ch. 1, Sect. 4, Theorem 1]), which assumes a slower growing normalising sequence for a proper limit in the law of large numbers. As shown in Dvoretzky and Erd˝os [8, Theorem 3] for a simple random walk and in ˇ Cerný [3] for a general one with zero drift and finite covariance matrix, it is n/ log n in two dimensions. In present article, we show that the law of large numbers for L n (α) with a non-zero limit and normalising sequence n holds in any dimension d for any transient random walk, that is, when the probability of its return to the origin, γ := P{Sn = 0 for all n ≥ 1}, is strictly positive, γ > 0. We assume in addition that γ < 1 which excludes a trivial case where either l(n, x) = 1 or 0 for all
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