Optimal Bounds for the Variance of Self-Intersection Local Times

Abstract

For a -valued random walk , let be its local time at the site . For , define the -fold self-intersection local time as . Also let be the corresponding quantities for the simple random walk in . Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely -dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, . In particular, for any genuinely -dimensional random walk, with , we have . On the other hand, in dimensions we show that if the behaviour resembles that of simple random walk, in the sense that , then the increments of the random walk must have zero mean and finite second moment.

1. Introduction and Main Results

Let be independent, identically distributed, -valued random variables, and define the random walk , , for . The special case with , for all with , is known as the simple random walk in and will be denoted by .

Let be the local time of at the site , and define for a positive integer the -fold self-intersection local time We will denote the corresponding quantities for simple random walk in by or simply when the dimension is clear from the context.

Let and be, respectively, the semigroup and the group generated by the support of , Following Spitzer [1], we call the random variable and the random walk it generates genuinely -dimensional if the group is -dimensional.

The quantity has received considerable attention in the literature due to its relation to self-avoiding walks and random walks in random scenery. In particular let the random scenery be a collection of i.i.d. random variables, independent of , and define the process , Then is commonly referred to as random walk in random scenery and was introduced in Kesten and Spitzer [2], where functional limit theorems were obtained for under appropriate normalization for the case . The case , with centered with nonsingular covariance matrix, was treated in [3] where it was shown that converges weakly to Brownian motion. As is obvious from the identities and , limit theorems for usually require asymptotic results for the local times of the random walk .

Such asymptotic results are usually obtained from Fourier techniques applied to the characteristic function , under the additional assumption of a Taylor expansion of the form , where is a positive definite covariance matrix [3–7], which further requires that and . Similar restrictions are also required for the application of local limit theorems such as in [8, 9].

In this paper, motivated by the results of Spitzer [1] for genuinely -dimensional random walks and the approach of Becker and König [10], we will study the asymptotic behavior of without imposing any moment assumptions on the random walk. The central idea behind our approach is to compare the self-intersection local times of a general -dimensional walk with those of its symmetrised version. In addition we will compare the self-intersection local times of a general -dimensional random walk with those of the -dimensional simple symmetric random walk, . It is well known that, for some positive constants , as , for Several other cases have been treated in the literature, using a variety of methods.

A careful look at the literature reveals that the most difficult case in is the near transient recurrent case, where , which corresponds to genuinely -dimensional symmetric recurrent random walks, which will be referred to as a critical case. Surprisingly enough, the variance of the self-intersection local times in the critical case is asymptotically the largest.

Theorem 1. Let be independent, identically distributed, and genuinely -dimensional -valued random variables, for any . Then, there exist positive constants , depending on and the distribution of , such that for all large enough

The result was motivated by [1, 10] and improves related results of Becker and König for and . Several cases treated in [3, 4, 10–13] can then be obtained as particular cases.

Moreover, we also show the surprising converse. More precisely, we show that the right asymptotic behaviour of implies that the jumps must have zero mean and finite second moment.

Theorem 2. Let , be independent, identically distributed, and genuinely -dimensional with . If then and .

As it follows from Theorem 3 given below for and from Theorem in Chen [12] for , if and , then .

For any genuinely -dimensional random walk with finite second moments and zero mean, the asymptotic behaviour of is similar to that of the -dimensional simple symmetric random walk. Also, as it follows from our general bounds (see Proposition 4 and Corollary 7) that the asymptotic results for the genuinely -dimensional random walk can be reproduced by those of the symmetric one-dimensional random walk with appropriately chosen heavy tails, as was indicated by Kesten and Spitzer [2]. The proofs are based on adapting the Tauberian approach developed in [13].

Theorem 3. Let , and suppose that for one haswhere is a nonsingular covariance matrix and for and for as . Then where and and are defined in (58) and (63), respectively.
Moreover, if is the self-intersection local time of another random walk, independent of , whose characteristic function also satisfies (6), then .

2. Proofs

2.1. General Bounds

We first develop a technique to treat random walks with independent but not necessarily identically distributed increments.

Proposition 4 (general upper bound). Assume that are independent -valued random variables and let . Suppose further that for all and integers , with and any , one haswhere is nonincreasing and is nonincreasing in and is nondecreasing and subadditive in in the sense that , for some constant independent of , , and . Then, for some constant depending only on

Proof of Proposition 4. We first write out the variance as a sumAn important role is played by the manner in which the two sequences are interlaced, since, for example, if or , the term vanishes by the Markov property.
We will treat the sum over indices with . The sum over the remaining index set with can be treated in a similar fashion and will contribute a constant factor. Therefore, we assume that and we arrange the two sequences in an ordered sequence of combined length which we denote as ; we also define where if came from and if came from . Finally we define two new sequences , and , where , and , for . Notice that since we assume that , we have and . Let denote the interlacement index. The terms with vanish, while the terms with will be considered separately.
Terms with . We first consider the sum over the terms with for which we drop the negative part and obtain the boundSumming over the free index , it is clear that For any with , exactly elements are equal to 0, and therefore by Assumption with we have Letting denote an independent copy of the random walk and assuming without loss of generality that , we have that for any Let . Since is nonincreasing we have that and iterating this procedure, for , we have that . Combining the two bounds and summing over , we have that where is a constant depending only on .
Terms with . Next we consider the sum over the terms with , which occurs when, for some , the indices all lie in . Then it is easy to see that this sum is bounded above by

The following corollary provides explicit bounds in the cases that are usually considered in the literature.

Corollary 5. Assume that the conditions of Proposition 4 are satisfied with and . Then,

It is straightforward to see that Corollary 5 includes random walks with mean zero and finite second moment; for example, corresponds to and to . Therefore several relevant results in [3, 7–13] are obtained as a special case of Corollary 5 and extended to the case of independent but not necessarily identically distributed variables, for example, by applying the local limit theorem, as conducted in [8].

Also when the random walk increment is in the domain of attraction of the one-dimensional symmetric Cauchy law [13, 14] or in the case of planar random walk with second moments [3, 7–9, 11], it is well known that the conditions of Proposition 4 are satisfied with and .

However, we can do better for symmetric variables and show that condition implies , which together with the comparison technique motivates the following results. For a real number , we write for the integer part of .

Proposition 6 (bounds via comparison with characteristic function of symmetric random variables). Let , be independent -valued random variables and let . Assume that there exist a measurable function and a positive nonincreasing sequence , such thatfor all integers , all , and some positive constant . Then there exists another positive constant such that

Proof of Proposition 6. Using the notation of Proposition 4, for positive integers , with , , and any To find , notice that since , whence . Therefore, A telescoping argument implies that On the other hand for we can obtain a tighter bound through Combining the two bounds above it follows that is satisfied with . Thus all conditions of Proposition 4 are satisfied and the result follows.

The following corollary allows for the case where is regularly varying.

Corollary 7. Assume that the conditions of Proposition 6 are satisfied with , , where is slowly varying at . Then,

Several results in [3, 7–13] are obtained as a special case of Corollary 7 and can be extended to dependent variables, for example, a random walk driven by a hidden Markov chain. In addition, following [2], we can construct a one-dimensional symmetric random walk with characteristic function , where for and for , whose asymptotic behaviour is similar to that of genuinely -dimensional random walk.

The following example of genuinely -dimensional recurrent walk with infinite variance was motivated by Spitzer [1, pp. 87].

Example 8. Let be independent, identically distributed, -valued random variables, such that , for and . Let be the corresponding random walk in . Then we havefor . Under these assumptions we have that , which is in the critical range, where the random walk is recurrent, without second moment. To see why, we note that by a lengthy but straightforward calculation it can be shown that the characteristic function of satisfies (19) with The sequence is identified via Fourier inversion, polar coordinates, and a Laplace argument,

2.2. Bounds for Identically Distributed Variables

Proposition 9 (general upper bound for i.i.d.). Let , be independent, identically distributed, -valued random variables. Suppose that for any and all positive integers , , , and , with , it holds thatwhere is a nonincreasing sequence. Then for some constant we have that

Proof of Proposition 9. By inspecting the proof of Proposition 6, we notice that we only need to bound the term . Consider typical ordering and let us change variables to such that . Then the contribution to is given byWe keep fixed for and we sum over from to some . Then for given , the term in the sum is where . Then since , it is an easy exercise to show that this sum is bounded above by where . The result follows by summing over all indices apart from and changing the order of summation.

2.3. Proofs of Main Results

Proof of Theorem 1. We apply a comparison argument found to be useful in many areas (e.g., Montgomery-Smith and Pruss [15], and Lefèvre and Utev [16]). More specifically we bound the quantity by the corresponding quantity for the symmetrised random walk.
Following Spitzer’s argument we notice that with Since is the characteristic function of a symmetric random variable in , for some positive , we have , and, hence, The result follows from Proposition 9 applied with .

The proof of Theorem 2 will be based on the following lemma.

Lemma 10. Assume are independent, identically distributed, genuinely -dimensional random variables such that . Then there exists a monotone, slowly varying sequence , such that as and

Proof of Lemma 10. Without loss of generality we assume that is symmetric. Let . Following Spitzer, since is genuinely -dimensional, we may assume that there exist positive constants , such that for any unit vector we have that and for all . Let be the -dimensional Lebesgue measure on and the Lebesgue-Haar measure on . Notice that since , for any , we have .
Fix a small positive such that , and for any let . Then there exists small enough so that . We partition in two sets so that, for any direction , Hence, using -dimensional spherical coordinates, On the other hand, for any , Now, assume that . Then for any direction , by choice of and since is increasing in , for or , it must be the case that implying that, on the set , it must be that . Changing to -dimensional polar coordinates, we find that Overall, for , , and hence has Lebesgue measure .
Let be the cumulative distribution function of the random variable defined on the probability space with normalised Lebesgue measure. Then is continuous at and supported on . Moreover, we have that as . Therefore, for some positive sequence with , we have thatIt remains to show that there exists a positive, monotone, slowly varying sequence , such that as . Let and and for define recursively by , for , so that is monotone, implying that , and . Finally, take .

Proof of Theorem 2. Assume that and or . Then, by Lemma 10 there exists a slowly varying sequence as such that . Applying Corollary 7 with and we, respectively, find that Finally assume that and . Then whence it follows that (see, e.g., [17, Theorem 2.3.10]). Then inspecting the proof of Proposition 4, one can readily obtain the desired bound for the term, while with slight modification the bound for the term also follows.
Note that for the situation is much simpler since then and if or , .

Proof of Theorem 3. We first give the proof for the case . As in the proof of Proposition 4 we begin from expression (10) and define the sequences and for , and the quantity . Recall that measures the interlacement of the two sequences and . For example, occurs when either or , in which case the contribution vanishes by the Markov property. On the other hand when, for example, for some . Finally occurs when, for example, From the proof of Proposition 4, and using the bound , the terms of the sum are bounded above by , and thus the leading term appears when either , with other terms giving strictly lower order. We will therefore analyze these two situations in detail in order to derive the exact asymptotic constants. When , the two terms in the difference individually give the correct order and will be treated by the classical Tauberian theory. However for , the two terms only give the correct order when considered together. This however forbids the use of Karamata’s Tauberian theorem since the monotonicity restriction would require roughly that is symmetric. Thus the complex Tauberian approach, as developed in [13], is required to justify the answer.
Case  1 (). Assume that part of the sequence lies between and and the rest between and . Then using the change of variables we rewrite the positive term in (10) as Notice that from [13] we have that . Let Then, by direct calculations and Fourier inversion formula Next we consider the negative term in (10) By direct calculations and (6), and using Fourier inversion and (6) the internal sum behaves as Then, we have , whence the Tauberian theorem implies that . Most importantly we see that the lengths and locations of the chains, and , do not affect the asymptotic behaviour. Noting that if , , we can partition in ways, and thus overall the total contribution from terms with is Case  2 (). The typical term was introduced in (33) in the proof of Proposition 9. Now we let , with . By lengthy but direct calculations we can derive an expression of the form The approach developed in [13] can then be used to bound the error terms and show that
Finally taking into account the fact that can be in any of the intervals , for , the result follows the overall contribution of terms with The case for is very similar, so we move on to the case .
Case  3 ( and ). Using the same notation as before, we have three terms to consider , , and . We first consider . Letting and using the usual power series construction and spherical coordinatesand thus , where , where the answer can be justified following [13].
The term is trickier to compute. As usual we consider the power series Let be the cosine of the angle between and , which in spherical coordinates is Then as , using the expansion (6) The other integral is slightly easier and thus overall we must have thatwhence it follows that .
To prove the last claim let be another random walk, independent of , such that its characteristic function also satisfies the expansion (6). Then using [13, Lemma 3.1], one can adapt the proof of [13, Theorem 2.1] to show that .

Competing Interests

The authors declare that they have no competing interests.