The Mean Number of Sites Visited by a Random Walk Pinned at a Distant Point

This paper concerns the number Zn of sites visited up to time n by a random walk Sn having zero mean and moving on the two dimensional square lattice Z 2. Asymptotic evaluation of the conditional expectation of Zn for large n given that Sn = x is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in (x, n), |x| < cn.


Introduction and main results
This paper is a continuation of the paper [12] by the present author, where the expectation of the cardinality of the range of a pinned random walk is studied when the random walk of prescribed length is pinned at a point within a parabola of space-time variables.In this paper we deal with the case when it is outside a parabola at which the walk is pinned and compute the asymptotic form of the (conditional) expectation.To this end we derive a local limit theorem valid outside parabolas by using Cramér transform.
The random number, denoted by Z n , of the distinct sites visited by a random walk in the first n steps is one of typical characteristics or functionals of the random walk paths.The expectation of Z n may be regarded as the total heat emitted from a site at the origin which is kept at the unit temperature.The study of Z n is traced back to Dvoretzky and Erdös [2] in which the law of large numbers of Z n is obtained for simple random walk.
Nice exposition of their investigation and an extension of it is found in [10].For the pinned walk the expectation of Z n is computed by [12], [4].Corresponding problems for Brownian sausage have also been investigated (often earlier) (cf.[11], [3] for free motions and [6], [7], [14] for bridges).
Let S n = X 1 + • • • + X n be a random walk on the two-dimensional square lattice Z 2 starting at the origin.Here the increments X j are i.i.d.random variables defined on some probability space (Ω, F, P ) taking values in Z 2 .The random walk is supposed to be irreducible and having zero mean: E[X] = 0.Here and in what follows we write X for a random variable having the same law as X 1 . For * Tokyo Institute of Technology, Japan.E-mail: uchiyama@math.titech.ac.jp The mean number of sites visited and for µ ∈ R 2 let m(µ) be the value of λ determined by ∇φ(λ) Since ∇φ(0) = 0, if the interior of Ξ contains the origin, then so does the interior of ∇(Ξ).
Let f 0 (n) be the probability that the walk returns to the origin for the first time at the n-th step (n ≥ 1) and define Let Z n (n = 1, 2, . ..) denote the cardinality of the set of sites visited by the walk up to time n, namely Z n = {S 1 , S 2 , . . ., S n }.
Let Q be the covariance matrix of X and |Q| be the determinant of Q.
Theorem 1. Suppose that φ(λ) < ∞ in a neighborhood of the origin and let K be a compact set contained in the interior of Ξ.Then, and, uniformly for x ∈ Z 2 satisfying x/n ∈ ∇φ(K) and |x| ≥ √ n, as n → ∞. (1.3) Example 1.For symmetric simple random walk we have e φ(λ) = 1 2 cosh α + 1 2 cosh β for λ = (α, β).Given x/n = µ + o(1), the leading term nH(x/n) in (1.3) may be computed from The derivative of H along a circle centered at the origin directed counter-clockwise is given by , We see shortly that the behavior of the probability P 0 [S n = x] differs greatly in different directions of x as soon as |x|/n follows: for each a • > 0 it holds that uniformly for |x| < a where , where γ = 0.5772 . . .(Euler's constant).
Brownian analogue of (1.4) is given in [14], the proof being similar but rather more involved than for the random walk case.
Remark 1.By a standard argument we have Remark 2. For d ≥ 3 the results analogous to (1.4) are obtained by the same method.Here only a result of [12] for the case d = 3 is given: where q 0 = P [S n = 0 for all n ≥ 1].
Remark 3.For random walks of continuous time parameter the asymptotic form of the expectation are deduced from those of the embedded discrete time walks by virtue of the well-known purely analytic result as given in [5].
For the proof of Theorem 1 we derive a local limit theorem, an asymptotic evaluation of the probability P [S n = x], denoted by q n (x), for large n, that is sharp uniformly for the space-time region √ n ≤ |x| < εn (with some ε > 0) (Lemma 3).As a byproduct of it we obtain the following proposition which lucidly exhibits what happens for variables √ n < |x| << n with n large: if all the third moments vanish, then the ratio of the probabilities q n (x) among directions of x with the same modulus |x| can be unbounded as |x|/n 3/4 gets large; if not, this may occur as |x|/n 2/3 gets large.This result though not directly used in the proof of Theorem 1 is interesting by itself.Proposition 2. Uniformly in x, as n → ∞ and |x|/n → 0, ] and κ 4 is a homogeneous polynomial of degree 4. If all the third moments of X vanish, then ECP 20 (2015), paper 17.
for x = (x 1 , x 2 ) ∈ Z 2 with n + x 1 + x 2 even.This formula, however, can be obtained rather directly if one notices that in the frame obtained by rotating the original one by a right angle the two components in the new frame are symmetric simple random walks on Z/ √ 2 that are independent of each other and use an expansion of transition probability of these walks as given in [8] (Section VII.6, problem 14).
The arguments involved in this subsection partly prepares for the proof of (1.3).
By definition λ = m(µ) is the inverse function of The Taylor expansion of φ about the origin up to the thid order is given by hence for |µ| small enough, ). (2.3) Here Q(λ) = λ • Qλ, the quadratic form determined by the matrix Q and similarly Now we compute H(µ) by using (1.5).From (2.3) and φ(iθ Substitution into (1.5) and a simple computation show

A local limit theorem.
Let q(x) denote the probability law of the increment of the walk: q(x) = P [X = x].Let µ = ∇φ(λ) with λ in the interior of Ξ and define ECP 20 (2015), paper 17.
Let q n and q n µ be the n-fold convolution of q and q µ , respectively.Then ) n e −m(µ)•x q n µ (x). (2.4) Let Q µ denote the covariance matrix of the probability q µ and Q −1 µ (x) the quadratic form determined by Q −1 µ .Lemma 3. Let K be a compact set contained in the interior of Ξ (as in Theorem 1).Then uniformly for y ∈ Z 2 − nµ and for µ ∈ ∇φ(K), as n → ∞ Here N may be an arbitrary positive integer, ν is the period of the walk S n , 1(S) is 1 or 0 according as the statement S is true or false, σ 2 µ denotes the square root of the determinant of Q µ and where P µ j is a polynomial of degree at most 3j determined by the moments of q n µ and odd for odd j.
Proof.This lemma may be a standard result.In fact it is reduced to the usual local central limit theorem as follows.Let ψ µ (θ) be the characteristic function of q µ and put ψµ (θ) = x q µ (x)e iθ•(x−µ) , so that where T = [−π, pi) × [−π, pi).Since ∇ ψµ (0) = 0, the Hessian matrix of ψµ at zero equals Q µ and p µ (x)|x| 2N < ∞ for all N > 0, the usual procedure to derive the local limit theorem (see [9]; also Appendix (A) for the case ν > 1 if necessary) shows that the right-most member equals that of the formula of the lemma.
In the formula of Lemma 3 the trivial factor 1(q n (nµ + y) = 0) may be replaced by 1(nµ + y ∈ Λ n ); also, for each k ∈ Z, q n µ (nµ + y) may be replaced by q n µ ((n − k)µ + y), hence by q n+k µ (nµ + y).Thus we can reformulate Lemma 3 as in the following Corollary 4. Let K be a compact set contained in the interior of Ξ.Then for each k ∈ Z, uniformly for y ∈ Z 2 − nµ and for µ ∈ ∇φ(K), as n → ∞ with the same notation as in Lemma 3.
The mean number of sites visited Proof of Proposition 2. In Lemma 3 we take µ = x/n.It follows that with λ = m(µ) In view of (2.4) and Lemma 3 we have only to compute asymptotic form of By (2.1) and (2.3) for |µ| small enough, where κ 4 (µ) is a polynomial of degree 4.
Assume that all the third moments of X vanish.Then, in place of (2.1) and (2.2) we have (2.7) respectively.Substituting these formulae into m(µ) • µ − φ(m(µ)) we observe that the term involving b(µ) disappears from the fourth order term by cancellation and hence that in which we find the explicit form of κ 4 (µ) as presented in the proposition.
Thus we have proved (1.3) and hence Theorem 1.

Appendix
(A) In the case when the period ν is larger than 1 the evaluation of the integral in (2.5) is reduced to that for the case ν = 1 by consideration of a property of its integrand that reflects the periodicity.By an elementary algebra one can find a point η ∈ R 2 that satisfies that for j = 0, 1, . . ., ν − 1, (Λ j is defined shortly after (2.6)).From this relation it follows that ψ(θ + 2πkη) = ψ(θ)e i2πk/ν (k = 0, . . ., ν − 1).

Now consider the expression q
and the right-hand sides equal [ψ(θ)] n e −ix•θ for all k if n − j equals zero in mod ν, while their sum over k vanishes otherwise.Choosing ε > 0 small enough, we may replace 2πη by a unique η k ∈ [−1 − ε, 1 + ε] such that η k − η ∈ Z 2 and apply the usual method for evaluation of Fourier integral.
Therefore, from the defining formula of H we have .
3/4gets large even if Q is isotropic.(See Proposition 2 below.)According to Theorem 1.2, in contrast to this, the leading term of E[Z n | S n = x] as x/n → 0 as well as that of H(µ) as µ → 0 is rotation invariant; only when |x|/n is bounded away from zero, E[Z n | S n = x] in general becomes dependent on directions of x.