Abstract
We introduce a natural family of random walks \(S_n\) on \(\mathbb{Z }\) that scale to fractional Brownian motion. The increments \(X_n := S_n - S_{n-1} \in \{\pm 1\}\) have the property that given \(\{ X_k : k < n \}\), the conditional law of \(X_n\) is that of \(X_{n - k_n}\), where \(k_n\) is sampled independently from a fixed law \(\mu \) on the positive integers. When \(\mu \) has a roughly power law decay (precisely, when \(\mu \) lies in the domain of attraction of an \(\alpha \)-stable subordinator, for \(0<\alpha <1/2\)) the walks scale to fractional Brownian motion with Hurst parameter \(\alpha + 1/2\). The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural “fractional” analogues of simple random walk on \(\mathbb{Z }\).
1 Introduction
Fractional Brownian motion is a one-parameter family of stochastic processes, mapping the real line to itself, that are the only stationary-increment Gaussian processes that, for some fixed \(H >0\), are invariant under the space-time rescalings \(S\rightarrow Y\) of the form \(Y_t = c^{-H} S_{c t}\) with \(c>0\). The parameter \(H\) is called the Hurst parameter, and may take any value in \((0,1)\). Fractional Brownian motion \(S^H:(0,\infty ) \rightarrow \mathbb{R }\) with Hurst parameter \(H\) satisfies
or, equivalently,
The process satisfies \(\mathbb{E }\big (S_t^H \big ) = 0\) for all \(t\in \mathbb{R }\). When \(H = 1/2\), it is Brownian motion, whose increments are of course independent. If \(H>1/2\), increments over two given disjoint intervals are positively correlated, while they have negative correlation if \(H < 1/2\).
Fractional Brownian motion was first considered by Kolmogorov [17] as a model of turbulence, and there is a now large literature treating this family of processes, e.g. as models in mathematical finance, and developing their stochastic calculus [20]. The stated characterization of fractional Brownian motion may permit this family of processes to be considered as at least slightly canonical. However, there are only a few examples of members of the family arising as a scaling or continuum limit of a discrete model (other than Brownian motion itself). The fractional Brownian motion with Hurst parameter \(1/4\) arises as a scaling limit for the tagged particle in a one-dimensional symmetric exclusion process [2]. For general values of \(H \in (0,1)\), fractional Brownian motion has been exhibited as a scaling limit of an average of a mixture of independent random walks, each walk having a decay rate for the correlation of its increments, which rate is selected independently from a law that depends on H [11]. In this paper, we present a simple discrete random walk that scales to fractional Brownian motion. The process may be considered to be a discrete counterpart to fractional Brownian motion.
In finance applications, as a model for the drift adjusted logarithm of an asset price, fractional Brownian motion retains much of the simplicity of ordinary Brownian motion (stationarity, continuity, Gaussian increments) but dispenses with independence of increments, thereby allowing for “momentum effects” (i.e., increment positive correlations), which have been observed empirically in some markets. One hypothesis is that momentum effects result from market inefficiencies associated with insider trading; if an event occurs of which the market is unaware, insider trading may cause the asset price to change gradually, over a period of time, instead of all at once—initially because of trades by individuals with privileged knowledge, later by a larger number of market participants. These ideas are discussed further in [14], where a model based on shocks of this form is shown to be arbitrage free and to have fractional Brownian motion as a scaling limit. (We will comment further on the possibility of arbitrage for fractional Brownian motion in Sect. 1.2). See also [3], which uses “investor inertia” to explain models in which drift-adjusted logarithmic price is a stochastic integral of fractional Brownian motion. Momentum effects appear naturally in the random walks we introduce: i.e., it will be easy to see why an “event” (the sampling of an increment of the walk at one point in time) has an influence on the expectation of future increments.
An informal description of the walk is as follows. Let \(\mu \) be a given law on the natural numbers (which we take to exclude zero). The walk associated with law \(\mu \) has increments, each of which is either \(-1\) or 1. Independent samples \(k_n\) of \(\mu \) are attached to the vertices of \(\mathbb{Z }\). The sequence of increments \(\big \{X_k: k \in \mathbb{N } \big \}\) is such that, given the values \(\big \{X_k : k < n \big \}, X_n\) is set equal to the increment \(X_{n - k_n}\) obtained by looking back \(k_n\) steps in the sequence. The walk is then defined by adding the successive increments from some fixed number. We remark that the notion of determining the value \(X_n\) of a process at time n by looking a random number of steps into the past also appears in some of the urn models and reinforced random walk models studied and surveyed by Pemantle in [21]. Recall that in the traditional Pólya’s urn process, the value of \(X_n\) represents the color of the nth ball added to an urn, and the conditional law of \(X_n\) given the past is that of a uniform sample from \(\{X_1, X_2, \ldots , X_{n-1}\}\). Our \(X_n\) differs from the Pólya’s urn process in that the past is infinite and the uniform sample is replaced by one having a power law distance in time from the present.
This description of the walk is a rough one, because it would require some initial condition to construct it. We will turn to a Gibbs measure formulation for a precise mathematical description. Fixing the law \(\mu \), we write \(G_{\mu }\) for the random directed spanning graph on \(\mathbb{Z }\) in which each vertex \(z\) has a unique outward pointing edge pointing to \(z - k_z\), where \(k_z\) is sampled independently according to the measure \(\mu \). We call \(z - k_z\) the parent of \(z\). The ancestral line of \(z\) is the decreasing sequence whose first element is \(z\) and each of whose terms is the parent of the previous one.
A law \(\lambda \) on functions mapping \(\mathbb{Z }\) to \(\{-1,1\}\) is said to be a \(\mu \)-Gibbs measure provided that, for any half-infinite interval \(R = \big \{x, \ldots , \infty \big \}\), the conditional law of \(\lambda \) on \(R\), given its values in \(R^c\), is given by sampling \(G_{\mu }\) and assigning to any element of \(R\) the value assigned to its most recent ancestor in \(R^c\). We will write \(X= X_{\lambda }: \mathbb{N } \rightarrow \{-1,1\}\) for a realization of the law \(\lambda \). (In general, a shift invariant probability measure on \(\{-1,1\}^{\mathbb{Z }}\) that has a specified conditional law for \(X_0\) given \(\{X_k: k < 0\}\) is called a g-measure for that specification [7, 15, 16]. One may view \(\mu \)-Gibbs measures as g-measures for a particular specification. We will not make use of this more general framework here).
A \(\mu \)-Gibbs measure is called extremal if it cannot be written as an average of two distinct \(\mu \)-Gibbs measures. There are at least two extremal \(\mu \)-Gibbs measures, whatever the choice of \(\mu \): those that assign unit mass to either the constant function equal to \(+1\) or to \(-1\). If \(\mu \) is such that the great common denominator of values in its support is 1, then the number of components in \(G_{\mu }\) is either equal almost surely to one or to \(\infty \). (We furnish a proof of this assertion in Lemma 2.1). If \(G_{\mu }\) has infinitely many components, then a one-parameter family \(\big \{ \lambda _p : p \in [0,1] \big \}\) of extremal Gibbs measures may be defined as follows. To sample from \(\lambda _p\), first sample \(G_\mu \) and then independently give each component of \(G_\mu \) a value of 1 (with probability \(p\)) or \(-1\) (with probability \(1-p\)). Then assign all of the vertices in that component the corresponding value.
Perhaps surprisingly, for such laws \(\mu \), any Gibbs measure is a mixture of these ones:
Proposition 1
Let \(\mu \) denote a probability measure on \(\mathbb{N } = \big \{1,2,\ldots \big \}\) the greatest common denominator of whose support is equal to one. If \(G_{\mu }\) has one component almost surely, then there are no extremal \(\mu \)-Gibbs measures other than the two trivial ones, \(\lambda _0\) and \(\lambda _1\). If \(G_{\mu }\) has infinitely many components almost surely, then the space of extremal Gibbs measures is the family \(\big \{ \lambda _p : p \in [0,1] \big \}\).
From this result, it is not hard to conclude that if the greatest common denominator of the support of \(\mu \) is some \(k \not = 1\), and \(\lambda \) is an extremal \(\mu \)-Gibbs measure, then the restriction of \(\lambda \) to points in \(k \mathbb{Z } +a\), for each \(a\in \{0,1,\ldots ,k-1\}\), will be an extremal \(\mu \)-Gibbs measure of the type described in Proposition 1, and that these measures will be independent for different values of a (though p may depend on a). Hence, there is no real loss of generality in restricting to the case that the greatest common denominator is 1, as we do in Proposition 1 (and throughout most of the remainder of this paper).
Next, we define the measures \(\mu \) that we will use for most of this paper.
Definition 1.1
Let \(\mu \) denote a probability measure on \(\mathbb{N }\). For \(\alpha \in (0,\infty )\), we say that \(\mu \in {\Gamma }_{\alpha }\) if there exists a slowly varying function \(L:(0,\infty ) \rightarrow (0,\infty )\) for which
for each \(n \in \mathbb{N }\). Recall that by slowly varying is meant
for any \(r > 0\).
Note that if we required L to be a constant function, then the measures satisfying the first condition would be simply those for which \(\mu \{n,\ldots ,\infty \}\) is a constant times \(n^{-\alpha }\). The generalization to slowly varying L is quite natural, for the following reason. Let \(R_\mu \) denote the random set \(\big \{\sum _{i=1}^j X_i : j \in \mathbb{N } \big \}\) of values assumed by partial sums of an independent sequence of samples \(X_i\) of the law \(\mu \). (Clearly, \(-1\) times the ancestral line of 0 has the same law as \(R_\mu \)). Then it turns out that when \(\alpha \in (0,1)\), the random set \(\epsilon R_\mu \) converges in law as \(\epsilon \rightarrow 0\) to the range of a stable subordinator with parameter \(\alpha \) if and only if \(\mu \in {\Gamma }_{\alpha }\) (Theorem 8.3.1 of [6]). Indeed, there is a sizable literature on probability distributions with power law decays up to a slowly varying function [6]. However, throughout this paper, when a result is stated for all \(\mu \in {\Gamma }_{\alpha }\), the reader may find it easier on a first reading to focus on the special case that L is constant.
Our next result relates the decay rate of the tail of \(\mu \) to the number of components in \(G_{\mu }\).
Proposition 2
Let \(\mu \in {\Gamma }_{\alpha }\) for some \(\alpha \in (0,\infty )\). If \(\alpha > 1/2\), then \(G_{\mu }\) almost surely has one component, while, if \(\alpha < 1/2\), then \(G_{\mu }\) almost surely has infinitely many.
To any \(\mu \)-Gibbs measure \(\lambda \), we associate a random walk \(S = S_{\lambda }: \mathbb{Z } \rightarrow \mathbb{Z }\) by setting \(S(0) = 0\) and \(S(n) = \sum _{i=1}^n X(i)\) for \(n > 0\), and \(S(n) = - \sum _{i = -n}^{-1} X(i)\) for \(n < 0\). We extend the domain of \(S\) to \(\mathbb{R }\) by linearly interpolating its values between successive integers.
We now ready to state our main result. Essentially, it says that, for S the random walk associated to an extremal \(\mu \)-Gibbs measure for a choice of \(\mu \in \Gamma _{\alpha }\), the time-scaled process \(S(nt)\), further rescaled by subtracting its mean and multiplying by a deterministic \(n\)-dependent factor, converges to fractional Brownian motion with Hurst parameter \(\alpha + 1/2\). This normalizing factor is written below as \(\tilde{c} n^{-\frac{1}{2} - \alpha } L(n)\). The explicit form for the constant \(\tilde{c}\) will be explained by the proof of Lemma 3.1, in which the asymptotic variance of \(S_n\) is determined.
Theorem 1.1
For \(\alpha \in (0,1/2)\), let \(\mu \in {\Gamma }_{\alpha }\). Let \(L:(0,\infty ) \rightarrow (0,\infty )\) be given by (1.1). Define \({\tilde{c}}>0\) by means of
where \(q_i = \mathbb{P } \big (i \in R_{\mu } \big )\) for \(i \ge 1\), and \(q_0 = 1\). Then, for each \(p \in (0,1)\), there exists a sequence of couplings \(\mathrm{C}_n\) of the process
and fractional Brownian motion \(S_{\alpha + 1/2}\) with Hurst parameter \(\alpha + 1/2\) such that, for each \(T > 0\) and \(\epsilon > 0\),
1.1 Discussion regarding Theorem 1.1
In light of Theorem 1.1, it is tempting to argue that the random walks we construct are in some sense the canonical fractional analogs of simple random walk on \(\mathbb{Z }\). To make this point, we note that any random walk on \(\mathbb{Z }\) (viewed as a graph) has increments in \(\{-1,1\}\), which means that its law, conditioned on \(\{X_n: n < M \}\) for some fixed M, is determined by the value of \(E[X_n | \{X_k: k<n\}]\). If we assume stationarity of increments and we further posit that \(E[X_0 | \{X_k: k<0\}]\) has a simple form—say, that it is a monotonically increasing linear function of \(\{X_k: k<0\}\)—then we have
for some \(p_i \ge 0\) with \(\sum p_i \le 1\). It is not hard to show that if \(\sum p_i < 1\), the process scales to ordinary Brownian motion. We are therefore left with the case \(\sum p_i = 1\), which corresponds to the walks we consider with \(\mu (\{i\}) = p_i\).
It seems plausible that any \(\mu \) for which the conclusion of the theorem holds (with the normalizing factor \(\tilde{c} n^{-\frac{1}{2} - \alpha } L(n)\) replaced by some deterministic function of \(n\)) must be a member of \(\Gamma _\alpha \). Similarly, it seems highly plausible (in light of (3.2)) that the closely related assertion that the variance of \(S_n\) is \(n^{2\alpha +1}\) (multiplied by a slowly varying function) implies \(\mu \in \Gamma _\alpha \). We will not prove either of these statements here.
We see that the model undergoes a phase transition at the value \(\alpha = \frac{1}{2}\). We remark that, while there are no non-trivial \(\mu \)-Gibbs measures for \(\mu \in {\Gamma }_{\alpha }\) with \(\alpha > 1/2\), there is nonetheless a further phase transition at the value \(\alpha = 1\), which is the maximal value for which an element \(\mu \in {\Gamma }_{\alpha }\) may have infinite mean. Indeed, suppose that we define the walk associated to a measure \(\mu \) by instead specifying its domain to be the positive integers and then making the following adjustment to the existing definition. As previously, we take \(\big \{k_n : n \in \mathbb{N } \big \}\) to be a sequence of independent samples of the measure \(\mu \), and, in the case that \(n - k_n \ge 0\), we continue to set the increment \(X_n\) to be equal to \(X_{n - k_n}\). In the other case, we choose this increment to be \(+1\) or \(-1\) with equal probability, independently of previous such choices. Then it is easily seen that the walk takes infinitely many steps of each type if and only if \(\mu \) has infinite mean.
Finally, we mention that the decomposition of \(\mathbb{Z }\) into components provided by Proposition 2 with a choice of \(\mu \in \Gamma _\alpha \) with \(\alpha < 1/2\) has something in common with the following process, discussed in [1]. A system of particles, one at each element of \(\mathbb{Z }\), are labelled, each by its location at an initial time \(t = 0\). Each pair of adjacent particles consider swapping locations at an independent Poisson sequence of times, but do so only if the higher-labelled particle lies on the right-hand-side in the pair just before the proposed swap. Each particle behaves as a second class particle in a totally asymmetric exclusion process, begun from an initial condition in which only sites to its left are occupied; as such, each has an asymptotic speed, which is a random variable having the uniform law on \([0,1]\). Defining a convoy to be the collection of locations inhabited at time zero by particles that share a common speed, ([1]) shows that the convoy containing the origin is almost surely infinite, has zero density, and, conditionally on the speed of the particle beginning at the origin, is a renewal process. As such, the collection of convoys provides a partition of \(\mathbb{Z }\) that has some similarities with \(G_{\mu }\), when it has infinitely many components.
1.2 Further comments
In financial contexts, where fractional Brownian motion may serve as a model of evolving prices or preference strengths, it is particularly natural to aim to simulate its future trajectory given some fixed past. The discrete model that we introduce is very well suited to this problem, because, given the past, \(N\) subsequent steps of the process may be sampled in time \(O(N)\) (neglecting logarithmic corrections arising due to random number generation). In comparison, fast Fourier transform methods for approximately sampling fractional Brownian motion necessarily take at least \(O(N \log N)\) time.
Mandelbrot and Van Ness introduced a representation of fractional Brownian as a weighted average of white noise. The representation of fractional Brownian motion stated in footnote 3 on page 424 of [19] inspires has a natural discrete counterpart, in which a weighted avarge of independent \(\pm 1\) coin tosses is used. However, the walks obtained in this way do not have \(\pm 1\) increments and are not as simple to generate as those of this paper. In regard to discretizations of fractional Brownian motion, we mention [23], that introduces a random walk approximation and shows its weak convergence to fractional Brownian motion.
The question of whether (and what kind of) arbitrage is admitted in financial models based on fractional Brownian motion or geometric fractional Brownian motion has been explored at some length in the finance literature. Fractional Brownian motion is not a semi-martingale: if the class of admissible trading strategies is large enough (and allows a potentially unbounded number of trades at arbitrarily small intervals) then one can indeed construct strategies that that admit positive return with probability one [22]. Even if such strategies involve trading that is too frequent to be practical, their existence complicates the construction of consistent derivative pricing models. A good deal of literature has addressed ways in which transaction costs, other strategy restrictions, or minor modifications to the model can be imposed to make the arbitrage opportunities go away. See [4] for a survey of this literature that clarifies how the presence or absence of arbitrage depends on the precise choice of admissible trading strategies. We note that in a simple discrete model such as ours—where the walk goes up or down by a unit increment at every discrete time step and all possible finite-length trajectories have positive probability—one cannot have arbitrage in the strong sense, so these particular subtleties are not relevant to us. It remains interesting, however, to think about the market efficiency implications of our model; the online counterpart [13] of this article contains some discussion on this matter in an appendix.
1.3 Structure of the paper
Section 2 is devoted to the analysis of \(\mu \)-Gibbs measures, and the proofs of Propositions 1 and 2 are presented there. In Sect. 3, we turn to the convergence of the discrete process to fractional Brownian motion, proving Proposition 3, which shows convergence in finite dimensional distributions of the rescaled walk to the continuous process. A useful tool here is an explicit asymptotic formula for the variance of the walk, which is presented in Lemma 3.1. In Sect. 4, we prove a correlation inequality applicable to any extremal Gibbs measures in Proposition 4, and then apply it to improve the topology of convergence to yield the \(L^{\infty }\)-small coupling given in Theorem 1.1.
The discrete processes that we introduce form a natural counterpart to fractional Brownian motion, and we are hopeful that they may serve as a tool for the analysis of the continuous process. They also appear to have various discrete relatives that are candidates for further enquiry. We take the opportunity to discuss this by presenting several open problems, in Sect. 5.
2 The space of Gibbs measures
We now prove the first two propositions. We begin with two simple lemmas.
Lemma 2.1
Let \(\mu \) be a measure on the positive integers, the greatest common denominator of its support being equal to one. Then \(G_{\mu }\) has either one component almost surely, or infinitely many almost surely.
Proof
We write \(A_i\) for the ancestral line of \(i \in \mathbb{Z }\). Let \(m_n = \mathbb{P } \big (A_0 \cap A_n \not = \emptyset \big )\). We now argue that if \(\inf _n m_n > 0\), then \(G_{\mu }\) has one component almost surely. If \(m : = \inf _n m_n > 0\), we may find \(g_n \in \mathbb{N }\) such that the ancestral lines from \(0\) and \(n\) meet at some integer exceeding \(- g_n\) with probability at least \(m/2\). If the ancestral lines \(A_0\) and \(A_n\) fail to meet in \(\{-g_n,\ldots \}\), then we sample them until each first passes \(-g_n\), at \(f_1\) and \(f_2\), say. Without loss of generality, \(f_1 < f_2\). Conditional on \(A_0 \cap A_n \cap \big \{ - g_n,\ldots \big \}= \emptyset \), and on \(f_1\) and \(f_2\), the probability that \(A_0 \cap A_n \cap \big \{ f_1 - g_{f_2 - f_1},\ldots ,f_1 \big \} \not = \emptyset \) is at least \(m/2\). Iterating, we construct a sequence of intervals each of which has a probability at least \(m/2\) of containing a point in \(A_0 \cap A_n\), conditionally on its precedessors not doing so. We see that the ancestral lines indeed meet almost surely.
If \(m_n \rightarrow 0\) subsequentially, then we will show that there exists a sequence \(\big \{ n_i: i \in \mathbb{N } \big \}\) such that
This suffices to show that \(G_\mu \) has infinitely many components almost surely, by the Borel–Cantelli lemma. To construct the sequence, for \(i \in \mathbb{N }\), set \(q_i = \mathbb{P } \big (0 \in A_i \big )\). Then, whenever \(n > l > 0\),
so that
It follows readily from \(\mathrm{g.c.d.} \mathrm{supp} (\mu ) = 1\) that \(q_j > 0\) for all sufficiently high \(j \in \mathbb{N }\). Supposing that we have constructed an increasing sequence \(\{ n_1,\ldots , n_{i-1} \}\) for which \(q_{n_j} > 0\) for \(j < i\), choose \(n_i > n_{i-1}\) satisfying \(q_{n_i} > 0\) and \(m_{n_i} \sum _{j=1}^{i-1} q_{n_j}^{-1} \le i^{-2}\). Note then that
the second inequality by (2.2). In this way, we construct a sequence satisfying (2.1).
\(\square \)
Proof of Proposition 1
The case that \(G_\mu \) has one component almost surely is trivial. Let \(\mu \) be a probability measure on \(\mathbb{N }\) whose support has greatest common denominator one, and for which \(G_{\mu }\) has infinitely many components almost surely. Let \(\lambda \) denote a \(\mu \)-Gibbs measure. For \(m \in \mathbb{Z }\), set \(\sigma _m = \sigma \big \{X_i: i < m \big \}\). By the backwards martingale convergence theorem (Sect. XI.15 of [8]),
exists \(\lambda \)-a.s. Note that it is an almost sure constant, because \(\lambda \) is extremal.
We will now argue that
We will denote the common value by p.
To show (2.4), let \(a,b \in \mathrm{supp}(\mu ), a \not = b\). Consider firstly the case that \(\vert b - a \vert \) divides \(\vert n -m \vert \). For \(k \in \mathbb{N }\), we enumerate the ancestral line \(A_k \,{=}\, \big \{ k \,{=}\, x_0(k),x_1(k),\ldots \big \}\) emanating from k in decreasing order. We define a coupling \(\Theta \) of \(A_n\) and \(A_m\). At any given step \(l \in \mathbb{N }\), the initial sequences \(\big \{x_0(n),\ldots , x_l(n) \big \}\) and \(\big \{ x_0(m),\ldots , x_l(m) \big \}\) have been formed. If \(x_l(m) \not = x_l(n)\), then we set \(x_{l+1}(m) - x_l(m) = -q\), where \(q\) has law \(\mu \). If \(q \not \in \{ a,b \}\), then we take \(x_{l+1}(n) - x_l(n) = -q\) also. If \(q \in \{ a,b \}\), then we take \(x_{l+1}(n) - x_l(n)\) to have the conditional distribution of a sample of \(\mu \) given that its value belongs to \(\{a,b\}\). If \(x_l(m) = x_l(n)\), we simply take \(x_{l+1}(m) - x_l(m) = x_{l+1}(n) - x_l(n) = -q\).
The sequence of differences \(\big \{ x_l(m) - x_l(n): l \in \mathbb{N }\big \}\) performs a random walk with independent jumps, a jump equal to zero with probability \(1 - 2 \mu \big \{a \big \} \mu \big \{b \big \}\), and, otherwise, with equal probability to be \(-\vert b - a \vert \) or \(\vert b - a \vert \), until the walk reaches 0. The walk beginning at a multiple of \(\vert b - a \vert \), the sequence of differences reaches zero, and then remains there, almost surely, by the recurrence of one-dimensional simple random walk.
If \(\vert b - a \vert \) does not divide \(\vert n -m \vert \), we begin by sampling initial segments of the two ancestral lines \(\big \{ x_0(n),\ldots , x_j(n) \big \}\) and \(\big \{ x_0(m),\ldots , x_j(m) \big \}\), where j is the stopping time
which is almost surely finite, because \(\mathrm{g.c.d.} \mathrm{supp} (\mu ) =1\). Indeed, this finiteness easily follows from the positivity of \(q_j\) for all sufficiently high \(j\). The coupling demonstrates that, given \(\epsilon > 0\), there exists \(l\) sufficiently negative that, \(\sigma _l\)-a.s.,
from which (2.4) follows.
We learn then that, for any \(n \in \mathbb{Z }\),
Let \(L \subseteq \mathbb{N }\) be finite. Set
For \(m,l \in \mathbb{Z }, m < l\), let \(v_{l,m} = \sup A_l \cap \big \{\ldots , m \big \}\) denote the first element in the ancestral line of \(l\) that is at most \(m\). For \(L \subseteq \mathbb{N }\) finite with \(m < \inf L\), write \(v_{L,m} = \big \{ v_{l,m}: l \in L \big \}\) for the locations of ancestors of elements of \(L\) that are at most \(m\) whose child is at least \(m+1\).
For \(\epsilon > 0\), set
Noting that \(\big \{ \big \vert v_{L,m} \big \vert : m < \inf L \big \}\) is a non-decreasing sequence that assumes a common value for all sufficiently high negative choices of \(m\), we find that \(g_{\epsilon ,L} > - \infty , \lambda \)-a.s.
For \(\epsilon > 0\) and \(j,k \in \mathbb{Z }, k < j\), write
Note that, for all \(\epsilon > 0\) and \(j \in \mathbb{N }\),
by (2.3) and (2.4). Given \(L \subseteq \mathbb{N }\) finite, set
so that \(j_{\epsilon ,L} > - \infty \) for each \(\epsilon > 0\) and finite \(L \subseteq \mathbb{N }\), by (2.5).
Let \(k \in \mathbb{Z }\) and \(\omega \in \{ -1,1 \}^{\{-\infty ,\ldots ,k\}}\). Define \(\big \{ Y_{k,l}^{(\omega )}: l \ge k + 1 \big \}\) by constructing an independent collection of ancestral lines (that do not coalesce on meeting) from the elements of \(\big \{ l \in \mathbb{Z }: l \ge k + 1 \big \}\), stopping each line on its arrival in \(\big \{ l \in \mathbb{Z }: l \le k \big \}\). For each \(l \ge k + 1\), we set \(Y_{k,l}^{(\omega )}\) equal to the \(\omega \)-value of the vertex at the end of the stopped ancestral line emanating from \(l\).
Let \(q_{\epsilon ,L} = j_{\epsilon ,v_{L,g_{\epsilon ,L}}}\). Set \(\Omega _{\epsilon }^* = \bigcap _{l \in v_{L,g_{\epsilon ,L}}} \Omega _{q_{\epsilon ,L},l}^{(\epsilon )}\), and note that \(\lambda \big (\Omega _\epsilon ^* \big ) \ge 1 - \vert L \vert \epsilon \).
Note further that, conditionally on \(v_{L,g_{\epsilon ,L}}\) and for any \(\omega \in \{-1, 1 \}^{\{ - \infty , \ldots , q_{\epsilon ,L}\}}\) for which
for each \(l \in v_{L,g_{\epsilon ,L}}\), the set of values \(\big \{Y_{q_{\epsilon ,L},l}: l \in v_{L,g_{\epsilon ,L}} \big \}\) are independent, each being equal to \(+1\) with probability at least \(p - \epsilon \) and at most \(p + \epsilon \).
The labelling \(\big \{X_i: i \in L \big \}\) is determined by the set of values \(\big \{X_i: i \in v_{L,g_{\epsilon ,L}} \big \}\). However, for each \(\omega \in \{-1,1\}^{\{\ldots ,q_{\epsilon ,L} \}}\) and for any finite set \(\sigma \subseteq \mathbb{Z }\) for which \(v_{L,g_{\epsilon ,L}} = \sigma \) is possible,
Note that the right-hand-side is at most \(\epsilon \), because \(v_{L,g_{\epsilon ,L}} = \sigma \) is possible. By taking \(\epsilon > 0\) arbitrarily small, we obtain the result. \(\square \)
Definition 2.1
Let \(\mu \) denote a probability distribution on \(\mathbb{N }\). Let \(R_{\mu }\) denote the random set \(\big \{ \sum _{i=1}^j X_i : j \in \mathbb{N } \big \}\) of values assumed by partial sums of an independent sequence of samples \(X_i\) of the law \(\mu \). When such a measure \(\mu \) has been specified, we will write \(p_n = \mu \{ n \}\) and \(q_n = \mathbb{P } \big (n \in R_{\mu } \big )\). We adopt the convention that \(q_0 = 1\).
Lemma 2.2
Let \(\alpha \in (0,1)\) and let \(\mu \in {\Gamma }_{\alpha }\).
-
(i)
We have that
$$\begin{aligned} \sum _{n=0}^{\lfloor x \rfloor } q_n \sim \frac{1 - \alpha }{\Gamma (2 - \alpha ) \Gamma (1 + \alpha )} x^{\alpha } L(x)^{-1}, \end{aligned}$$where \(f \sim g\) means \(\lim _{x \rightarrow \infty }{\frac{f(x)}{g(x)}} = 1\).
-
(ii)
Moreover, the sum
$$\begin{aligned} \sum _{n=0}^{\infty } q_n^2 \end{aligned}$$converges if \(\alpha \in (0,1/2)\) and diverges if \(\alpha \in (1/2,1)\).
Proof
Let \(p,q:[0,\infty ) \rightarrow [0,\infty ]\), given by
and
denote the Laplace transforms of these two sequences. We will analyse these transforms, (and, in later arguments, the Fourier transforms) of such sequences by means of some Tauberian theorems. A Tauberian theorem asserts that a sequence (in our case) has a given rate of decay if and only if its Laplace or Fourier transform has a corresponding asymptotic behaviour close to zero. Additional hypotheses, such as regular variation of the sequence (and, correspondingly, the transform), are required.
We will make use of two basic results concerning slowly varying functions. Firstly, the uniform convergence theorem (Theorem 1.2.1 of [6]) states that, if \(L:(0,\infty ) \rightarrow (0,\infty )\) is slowly varying, then
uniformly on each compact \(\lambda \)-set in \((0,\infty )\). Secondly, Potter’s theorem (Theorem 1.5.6 of [6]) states in part that, for any slowly varying function \(L:(0,\infty ) \rightarrow (0,\infty )\) that is bounded above and below on any compact interval in \((0,\infty )\), and for any \(\epsilon > 0\), there exists \(C = C_\epsilon \) such that, for all \(x,y > 0\),
From \(\big \{ p_n: n \in \mathbb{N } \big \} \in l_1\), the function \(p\) is differentiable on \((0,\infty )\), with
Setting \(U:(0,\infty ) \rightarrow (0,\infty )\) according to \(U(x) = \sum _{n=0}^{\lfloor x \rfloor }n p_n\), it follows from \(U(x) = \sum _{j=1}^{\lfloor x \rfloor } \sum _{i=j}^{\lfloor x \rfloor } p_i\), (1.1), (2.6) and (2.7) that \(U(x) \sim \frac{\alpha }{1-\alpha } x^{1-\alpha }L(x)\), \(x \rightarrow \infty \). A special case of Karamata’s Tauberian theorem (Theorem 1.7.1 of [6]) states that, if \(\big \{ a_n : n \in \mathbb{N }\big \}\) is a sequence of non-negative numbers, \(l:(0,\infty ) \rightarrow (0,\infty )\) is slowly varying, and \(c, \rho \ge 0\), then the following are equivalent:
Applying the theorem in the present case, we learn that
By (2.8), \(1 - p(\lambda ) \sim \frac{\Gamma (2-\alpha )}{1 - \alpha } \lambda ^\alpha L\big (1/\lambda \big ), \lambda \rightarrow 0^+\), since \(\int _0^\lambda t^{\alpha - 1} L \big (t^{-1} \big ) dt \sim \alpha ^{-1} \lambda ^\alpha L \big (\lambda ^{-1} \big )\), by means of (2.6) and (2.7). Noting that the Laplace transforms of the two sequences are related by \(q = \big (1 - p \big )^{-1}\), we learn that
The reverse implication in Karamata’s Tauberian theorem then yields the first statement of the lemma.
From (2.9) and
(the first inequality is Cauchy–Schwarz; the second inequality holds for \(\lambda > 0\) small enough), we find that \(\sum _{n=0}^{\infty } q_n^2\) diverges, if \(\alpha > 1/2\).
Set
and
to be the Fourier transforms of the two sequences.
From \(\mu \in \Gamma _\alpha \) with \(\alpha \in (0,1)\), we learn from Theorem 1 of [12] that
and
whence
if \(\mu (t,\infty ) = t^{-\alpha } L(t)\). Similarly,
From \(Q = (1-P)^{-1}\), we find that, for any constant \(C > \Gamma (1-\alpha )^{-1}\), there exists \(c > 0\) such that
for \(0 < t < c\), and, similarly,
From the support of \(\mu \) having greatest common denominator one, \(\vert P(\lambda ) \vert < 1\) for \(\lambda \in (0,2\pi )\). By the continuity of \(P:(0,\infty ) \rightarrow \mathbb{C }\),
for \(c > 0\) small.
By Parseval’s identity,
which, by (2.11), (2.12) and (2.13) is finite, provided that \(\alpha < 1/2\). \(\square \)
Proof of Proposition 2
Let \(\alpha > 0\) and \(\mu \in {\Gamma }_{\alpha }\). Let \(n,m \in \mathbb{N }\) satisfy \(n < m\). We wish to show that, if \(\alpha > 1/2\), then \(n\) and \(m\) lie in the same component of \(G_{\mu }\), while, in the case that \(\alpha < 1/2\), there is a positive probability (depending on \(m-n\)) that they lie in different components.
Let \(R^{(1)}_\mu \) and \(R^{(2)}_\mu \) be independent samples of \(R_{\mu }\). It is easily seen that \(n\) and \(m\) almost surely lie in the same component of \(G_{\mu }\) if and only if \(\mathbb{E } \big (\big \vert \big \{ n - R^{(1)}_{\mu } \big \} \cap \big \{ m - R^{(2)}_{\mu } \big \} \big \vert \big ) = \infty \). Moreover, if there exists some pair \((m,n) \in \mathbb{Z }^2\) for which this expectation is infinite, then clearly, it is infinite for all such pairs. Hence, there exists one component of \(G_{\mu }\) precisely when \(\mathbb{E }\big \vert R_{\mu }^1 \cap R_{\mu }^2 \big \vert \) is infinite. This expectation is equal to \(\sum _{n=1}^{\infty } q_n^2\), so the result follows from Lemma 2.2(ii). \(\square \)
3 Convergence to fractional Brownian motion
In this section, we establish that the walk has a fractional Brownian motion scaling limit in the sense of finite dimensional distributions:
Proposition 3
Let \(\mu \in {\Gamma }_{\alpha }\), for some \(\alpha \in (0,1/2)\). Recall the constant \({\tilde{c}}= {\tilde{c}}(p)\) for \(p \in (0,1)\) from (1.3). Then, for each \(p \in (0,1)\), the process \((0,\infty ) \rightarrow \mathbb{R }: t \rightarrow c n^{-\frac{1}{2} - \alpha } L(n) \big (S_{\lambda _p}\big (nt \big ) - n(2p -1)t \big )\) converges weakly (in the sense of finite dimensional distributions) as \(n \rightarrow \infty \) to fractional Brownian motion with Hurst parameter \(\alpha + 1/2\).
In preparation for the proof, let \(X: \mathbb{Z } \rightarrow \{-1,1\}\) be a sample of \(\lambda _p\), and let \(S: \mathbb{Z } \rightarrow \mathbb{Z }\) be the random walk satisfying \(S(0) = 0\) and \(S_n - S_{n-1} = X_n\) for each \(n \in \mathbb{Z }\). It is our aim to show that \(S_n\) is approximately normally distributed, when \(n\) is chosen to be high. We begin by finding an explicit expression for the variance of \(S_n\).
Lemma 3.1
We have that
where \(\big (\vert Q \vert ^2 \big )_i = \sum _{j=0}^\infty q_j q_{i+j}\) denotes the \(i\)-th Fourier coefficient of \(\vert Q \vert ^2\), and where
Proof
We begin by showing that
(In fact, (3.1) holds, more generally, for any \(\mu \) for which \(G_\mu \) has a.s. infinitely many components). To do so, write \(T_1,\ldots T_{r(n)}\) for the trees having non-empty intersection with \(\big \{ 1,\ldots , n \big \}\). Then
Note that, for \(i < j\),
and also that
since we adopt the convention that \(q_0 = 1\). Hence,
whence, (3.1). By (3.1), it suffices to show that
because Lemma 2.2(ii) implies that \(\big (\vert Q \vert ^2 \big )_0 < \infty \). To this end, note that, by (2.10) and \(Q = (1 - P)^{-1}\),
so that
By Theorem 4.10.1(a) of [6], using \(\vert Q(t) \vert ^2 \in \mathbb{R }\),
It follows that
Evaluating the integral, we obtain (3.2). \(\square \)
Proof of Proposition 3
There being a unique stationary Gaussian process \(S^H:\mathbb{R } \rightarrow \mathbb{R }\) with covariances given by \(\mathbb{E }\vert S_t^H \vert ^2 = t^{2\alpha + 1}\), it suffices, in light of Lemma 3.1, to establish that \(S_n\) has a distribution that is asymyptotically Gaussian. We will make use of the following notation. For \(M \in \mathbb{Z }\), set \(\sigma _M = \big \{ X_i:i< M \}\). Let \(X_n^M\) and \(S_n^M\) denote the expected values of \(X_n\) and \(S_n\) respectively given \(\sigma _M\). (Note that \(X_n^M\) is the expected value of \(X_k\) where \(k\) is the first element in the ancestral line of \(n\) that is less than \(M\)). Clearly, \(X_n^M\) and \(S_n^M\) are martingales in \(M,\) and \(S_n^M = S_n\) when \(n \ge M\). We will establish that \(S_n\) has an asymptotically Gaussian law by applying the martingale central limit theorem to \(S_n^M\). This will require showing that \(S_n^M\) has small increments compared to its total size and a use of a long-range near-independence argument to show that the sum of the conditional variances of the increments is concentrated. We now state the martingale central limit theorem in the form that we require. See Theorem 7.2 in Chapter 7 of [10], and the remark following its proof, for a derivation.
Definition 3.1
We say that \(X_{n,m}, \mathbb{F }_{n,m}, n \ge 1, 1 \le m \le k_n\) is a martingale difference array if \(X_{n,m} \in \mathbb{F }_{n,m}\) and \(\mathbb{E }\big (X_{n,m} \big \vert \mathbb{F }_{n,m-1} \big ) = 0\) for \(1 \le m \le k_n\). Let \(V_n = \sum _{i=1}^{k_n}\mathbb{E }\big (X_{n,m}^2 \big \vert \mathbb{F }_{n,m-1} \big )\).
Theorem 3.1
Suppose that \(\big \{ X_{n,m} , \mathbb{F }_{n,m} \big \}\) is a martingale difference array. Let \(X_{n,0} \in \mathbb{F }_{n,0}, n \in \mathbb{N }\). Set \(S_n = \sum _{j=0}^{k_n} X_{n,j}\). Assume that, for some sequence \(\epsilon _n \rightarrow 0\),
-
\(\big (\mathbb{E }V_n \big )^{-1} \big \vert X_{n,m} \big \vert ^2 \le \epsilon _n \, \, \mathrm{for all }\,m \in \{1,\ldots , k_n\}\)
-
\(\big (\mathbb{E }V_n \big )^{-1} \mathrm{Var} \big (X_{n,0} \big ) \le \epsilon _n\) for \(n\) sufficiently high,
-
\(\frac{V_n}{\mathbb{E }V_n} \rightarrow 1, \, \, \mathrm{in probability }\).
Then \(\mathbb{E }(V_n)^{-1/2} S_n\) converges in distribution to the normal distribution of mean zero and unit variance.
We will apply the result in the following way. Let \(\big \{\epsilon _n: n \in \mathbb{N }\big \}\) be a given sequence converging to zero. Let \(\big \{ k_n: n \in \mathbb{N }\big \}\) be a sequence satisfying
where the constant \(K_\alpha \) is specified in Lemma 3.1. We in addition assume that \(k_n \ge Cn\) for each \(n \in \mathbb{N }\), and for some large constant \(C > 0\). We choose as martingale difference array, \(X_{n,0} = S_n^{-k_n}, \mathbb{F }_{n,0} = \sigma _{- k_n}\) for each \(n \ge 1\), and \(X_{n,i} = S_n^{-k_n + i} - S_n^{-k_n + i - 1}, \mathbb{F }_{n,i} = \sigma _{- k_n + i}\) for each \(n \ge 1\) and \(1 \le i \le k_n + n\).
Clearly, we must verify that the hypotheses of Theorem 3.1 hold. The first column of the martingale difference array has been chosen to meet the second condition listed in the theorem, as we now confirm. Note that
By (3.3) and Lemma 3.1, we see that
Using \(\mathrm{Var} (S_n) = \mathrm{Var}(S_n - S_n^{-k_n}) + \mathrm{Var} (S_n^{-k_n})\) and (3.4), we obtain \(\mathrm{Var} S_n^{-k_n} \le \epsilon _n \mathbb{E }V_n\) for \(n\) sufficiently high, which is indeed the second hypothesis of Theorem 3.1.
The general martingale difference term takes the form \(S_n^m - S_n^{m-1}\), for some \(n \ge 1\) and \(m \le n\). A basic observation is that, for any \(M < n\),
where \(q_i\) (as in Definition 2.1) denotes the probability that the vertex \(i\) has 0 as an ancestor. As stated earlier, \(X_n^M\) is the expected value of the value observed by tracing back the ancestral line in a sample of \(G_{\mu }\) from \(n\), until an ancestor with index strictly less than \(M\) is reached. The difference in the above equation, then, is due to the event that the ancestral line reaches \(M\).
If \(n \ge 1\) and \(m < 0\), note that \(S_n^{m+1} - S_n^m = \sum _{i=1}^n \big (X_i^{m+1} - X_i^m \big )\), so that (3.6) yields
where we define \(F_n^m = \sum _{i=1}^n q_{i - m}\). In the case where \(n \ge 1\) and \(m \in \{0,\ldots ,n-1\}\), on the other hand,
by the convention that \(q_0 = 1\). We record this as
Note also that, for any \(m \in \mathbb{Z }\),
where \(P_m\) is defined to be \(\mathbb{P } \big (X_m = 1 \big \vert \sigma _m \big )\) and may be written \(\big (X_m^m + 1 \big )/2\). The definition of \(V_n\) and the expressions (3.7) and (3.8) now give the formula
which may be written
As such, the following lemma shows that the third hypothesis of Theorem 3.1 is satisfied in the present case.
Lemma 3.2
Setting \(c_0 = 4 \mathbb{E }\big (P_M (1-P_m) \big )\),
where \(E_1(n) \rightarrow 0\) in probability. We have further that
where \(E_2(n) \rightarrow 0\) in probability.
The ergodic theorem might be used to prove a concentration inequality of this sort. We prefer, however, to derive it directly, from a second moment estimate. We begin by observing the following.
Lemma 3.3
Set
and write
Then, for each \(\epsilon > 0\), there exists \(K_0 \in \mathbb{N }\) such that \(\vert l - m \vert \ge K_0\) implies that \(\vert \rho _{l,m} \vert < \epsilon \).
Proof
For \(i \in \mathbb{Z }\) and \(k \in \mathbb{N }\), set \(X_{i,k}^i = \sum _{j=1}^k \mu \{ j \} X_{i-j}\). Note that
For \(i,j \in \mathbb{Z }\) and \(k \in \mathbb{N }\), set
where recall that we write \(A_n\) for the ancestral line of \(n \in \mathbb{Z }\). Shortly, we will show that, for each \(\epsilon > 0\) and \(k \in \mathbb{N }\), there exists \(n_0 \in \mathbb{N }\) such that, if \(i,j \in \mathbb{Z }\) satisfy \(\vert i - j \vert \ge n_0\), then
For now, we show that this suffices for the proof of the lemma.
Note that, for any \(i,j \in \mathbb{Z }\) and \(k \in \mathbb{N }\) for which \(\vert i -j \vert \ge k\), if \(Y \in \sigma \big \{ X_{i-k},\ldots , X_{i-1} \big \}\) and \(Z \in \sigma \big \{ X_{j-k},\ldots , X_{j-1} \big \}\)
Note that
the second equality by (3.10), the third by \(\vert i - j \vert \ge n_0\) and (3.11), the fourth by (3.12), and the sixth by (3.10) and (3.11). Thus, for any \(\epsilon > 0\), there exists \(n_0 \in \mathbb{N }\) such that, if \(i,j \in \mathbb{Z }\) satisfy \(\vert i - j \vert \ge n_0\), then
This completes the proof of the lemma, subject to verifying (3.11). Note that, for this, it suffices to show that, for any \(\epsilon > 0\), there exists \(n_0 \in \mathbb{N }\) such that, if \(i,j \in \mathbb{Z }\) satisfy \(\vert i - j \vert \ge n_0\), then \(\lambda \big (A_i \cap A_j \not = \emptyset \big ) < \epsilon \). To see this, note that, for \(i < j\),
so that Lemma 2.2(ii) yields the desired conclusion. \(\square \)
Proof of Lemma 3.2
To verify the first claim, we aim to show that
where here we write
We will require
as \(n \rightarrow \infty \).
Indeed, by computing the second moment of \(Y_n\), (3.13) follows from (3.14) by means of Lemma 3.3.
We turn to proving (3.14). Recall that we supposed that \(k_n \ge Cn\), with \(C > 0\) a large constant. Note that
with \(c > 0\) a constant satisfying \(c < \Big (\frac{2(1-\alpha )}{\Gamma (2 - \alpha ) \Gamma (1 + \alpha )} \Big )^2\), and the latter inequality by Lemma 2.2(i), (2.6) and (2.7). For \(0 \le m \le \lfloor Cn \rfloor \) and a constant \(C_0 > \frac{2(1-\alpha )}{\Gamma (2 - \alpha ) \Gamma (1 + \alpha )}\)
by Lemma 2.2(i) and (2.6). We learn that
To treat the bound on \(F_M^n\) for \(- k_n < M < - n\), note that, if \(m > n\), then either
or
the first alternative holding provided that at least one half of the sum \(\sum _{i=1}^n q_{i+m}\) is contained in its first \(\lfloor n/2 \rfloor \) terms. From this, we conclude that, for any \(M < -n\),
whence
by increasing the value of \(k_n \in \mathbb{N }\) if necessary. This completes the proof of (3.14).
We have verified the first claim of the lemma, since it evidently follows from (3.13), subject to proving the lemma that follows.
The second claim is proved analogously. The analog of (3.15) is
which follows from the inequality
which is a consequence of Lemma 2.2(i), and the bound
which follows from Lemma 2.2(i) and (2.6). \(\square \)
It remains to show that the first hypothesis of Theorem 3.1 holds. Recalling (3.7), we see that, for \(- k_n < M < 0, \big \vert S_n^{M+1} - S_n^M \big \vert ^2 \le 4 \big (F_M^n \big )^2\), whose right-hand-side is \(o\big (\mathbb{E }V_n \big )\) by (3.14), (3.9) and Lemma 3.2. For \(0 \le m < n\), we use (3.8) to find that \(\big \vert S_n^{m+1} - S_n^m \big \vert ^2 \le 4 \big (\sum _{l=1}^{n-m} q_l \big )^2\). However, this right-hand-side is at most \(4 c n^{2\alpha } L(n)^{-2}\) by Lemma 2.2(i). From the discussion after (3.5), we know that \(\mathbb{E }V_n \ge \big (1-\epsilon _n \big ) \mathrm{Var} S_n\) for \(n\) sufficiently high, so that \(4 c n^{2\alpha } L(n)^{-2} \le O \big (n^{-1} \mathbb{E }(V_n) \big )\) by Lemma 3.1. Thus, the first hypothesis of Theorem 3.1 is indeed satisfied. \(\square \)
4 The FKG inequality and convergence in \(L^{\infty }\)
This section is devoted to the remaining step in the proof of Theorem 1.1, namely, to improving the topology in our convergence result.
Proposition 4
Let \(\mu \) be any probability measure on \(\mathbb{N }\), and let \(\lambda \) be any extremal \(\mu \)-Gibbs measure. Let \(X,Y \in L^2 \big (\lambda \big )\) be two increasing functions, where \(\{ -1,1 \}^{\mathbb{Z }}\) is given the natural poset structure. Then \(X\) and \(Y\) are not negatively correlated under \(\lambda \).
Proof
Note that \(\mathrm{Cov} \big (X,Y \big ) \ge 0\) if and only if
For any \(Z \in L^2 \big (\lambda \big )\), we write
where \(\sigma _n\) is the \(\sigma \)-algebra generated by coordinates with index strictly less than \(n\). In regard to (4.2), note that the maps \(L^2(\lambda ) \rightarrow \mathbb{R }: Z \rightarrow \mathbb{E }(Z\vert \sigma _n)\) that project functions onto their mean given \(\sigma _n\) form an increasing sequence of projections in the Hilbert space \(L^2(\lambda )\). The identity (4.2) expresses Z as a sum of differences of the successive projections. As such, this is a sum of orthogonal vectors in \(L^2(\lambda )\). The sum of a countable collection of orthogonal vectors in a Hilbert space has finite norm if and only if the sum of the squares of the norms of these vectors is finite. The norm of the sum has a square given by the sum of the squares of the norms of the constitutent vectors. From this, we obtain
We find then that
Note that, in the summand on the right-hand-side of (4.3), \(\mathrm{Var}\) denotes a conditional variance: the data in \(\sigma _{n-1}\) is fixed, and the relevant randomness arises from the bit with index \(n-1\).
Given \(\sigma _{n-1}\), each of \(X\) and \(Y\) does not decrease if we condition on the value of the \(n\)-th coordinate to be \(1\), and does not increase if this value is conditioned to be \(-1\). Thus,
\(\sigma _{n-1}\)-a.s. Taking expectation over \(\sigma _{n-1}\) and summing yields (4.1) by means of (4.3). \(\square \)
Definition 4.1
Recall that \(S_0 = 0, S_i = \sum _{j=1}^i X_j\) if \(i > 0\), and \(S_i = - \sum _{j=i}^{-1} X_j\) if \(i < 0\), where \(X_i = X_n{\lambda _p}(i)\). For \(l < m\), write
and
Lemma 4.1
There exists a constant \(C_0 > 0\) such that, for each \(\epsilon > 0\), there exists \(n_0 = n_0(\epsilon )\) such that, for \(n \ge n_0\),
where the constant \({\tilde{c}}= {\tilde{c}}(p)\) was defined in (1.3).
Proof
We write \(A_n = A_{0,n}\) and \(B_n = B_{0,n}\). For \(n \in \mathbb{N }\), set \(h(n)\! =\! 3 {\tilde{c}}^{-1} \sqrt{\log \big (\epsilon ^{-1} \big )} n^{1/2 + \alpha } L(n)^{-1}\). We begin by establishing the following statements. For any \(\epsilon > 0\), there exists \(n_0 = n_0(\epsilon )\) such that, if \(n \ge n_0(\epsilon )\), then, for any \(K > 0\),
and
Note that \(A_n\) is an increasing and \(B_n\) a decreasing random variable. As such, supposing that \(\mathbb{P } \big (B_n > K + h(n) \big ) \ge \epsilon \) and \(\mathbb{P } \big (A_n \le K \big ) \ge \epsilon \), then
the first inequality by means of Proposition 4. However, \(A_n - B_n = S_n - (2p-1)n\), so that
the inequality valid for \(n\) sufficiently high, because Proposition 3 implies that \(n^{-1/2 - \alpha } L(n) {\tilde{c}}\big (S_n - (2p-1)n \big )\) converges in distribution to a Gaussian random variable of mean zero and unit variance. In this way, we establish (4.4), with (4.5) following similarly.
We now show that there exists \(\alpha _n \in (0,\infty )\) such that, for \(\epsilon > 0\) and \(n \ge n_0(\epsilon )\),
and
To this end, set
Note that \(\mathbb{P } \big (A_n > K(n) \big ) \ge \epsilon \) and \(\mathbb{P } \big (A_n > K(n) + \epsilon \big ) < \epsilon \). The first inequality forces \(\mathbb{P } \big (B_n > K(n) - h(n) \big ) \ge 1 - \epsilon \) by means of (4.5), which gives
by (4.4). We now set \(\alpha _n = K(n) - h(n)\) and note that, if \(\epsilon < h(n)\), then (4.6) holds. Indeed, these choices for \(\alpha _n\) and \(\epsilon \), alongside \(P(A_n > K(n) + \epsilon ) < \epsilon \), yield
The bounds (4.9) and (4.10) imply that
whence (4.6). Note now that \(\mathbb{P } \big (B_n \ge \alpha _n + 2h(n) + \epsilon \big ) < \epsilon \): for otherwise, (4.4) would imply that \(\mathbb{P } \big (A_n > \alpha _n + h(n) + \epsilon \big ) \ge 1 - \epsilon \), contradicting our assumption. We see that
whence (4.7).
We set \(r_n = \alpha _n n^{-1/2 - \alpha } L(n)\). We claim that, for any \(K \in \mathbb{N }\), there exists \(n_0 \in \mathbb{N }\) such that \(n \ge n_0\) implies
From this, we will argue that
for some \(C_0 > 0\). Indeed, fixing \(K\in \mathbb{N }\) in (4.11) high enough that \(2 K^{-1/2 - \alpha } < 1\), we find that the limsup of \(r_m\) along the subsequence \(\big \{ m = Kn: n \in \mathbb{N }\big \}\) is at most \((20 + 12 + 1) \big ({\tilde{c}}^{-1} \sqrt{\log \big (\epsilon ^{-1} \big )} \big )\). Note then that \(\big \{ A_n: n \in \mathbb{N }\big \}\) is a random process whose increments are in absolute value at most one. Thus, the definition (4.8) implies that \(\vert \alpha _m - \alpha _n \vert \le \vert m - n \vert \). From this, we obtain the above bound on the limsup of \(r_m\) over all \(m \in \mathbb{N }\) (with an arbitrarily small addition to the value 33); and thus (4.12).
In tandem with (4.6), (4.12) yields
from which the statement of the lemma follows. To derive (4.11), note that
By the convergence for finite-dimensional distributions stated in Proposition 3, we have that, for each \(\epsilon > 0\), there exists \(n_0 = n_0(\epsilon )\) such that, for \(n \ge n_0\),
Note also that
for \(n \ge n_0(\epsilon )\). Choosing \(\epsilon > 0\) to satisfy \((3K-1) \epsilon \le 1 - 2 \epsilon \), from (4.14) and the inequality arising from (4.6) by the substitution of \(Kn\) for \(n\), we see that
Rearranging, and by \(L:(0,\infty ) \rightarrow (0,\infty )\) being a slowly varying function, we obtain (4.11). \(\square \)
Proof of Theorem 1.1
We will make use of:
Lemma 4.2
There exists a constant \(C \in (0,\infty )\) such that, for any \(\epsilon _0 > 0\) sufficiently small and for \(\delta > 0\),
Remark
An analogous bound holds for the counterpart of \(A_{l,m}\), in which the maximum taken in the definition is replaced by a minimum.
Proof
Note that (2.7) implies that, for n sufficiently high,
The probability that we must estimate is bounded above by
Setting \(\epsilon > 0\) in Lemma 4.1 according to \( C_0 + 50 {\tilde{c}}^{-1} \sqrt{\log \big (\epsilon ^{-1} \big )} = \delta \epsilon _0^{-1/4 - \alpha }\), we obtain that the last displayed expression is at most
for some constant \(C>0\), for \(\epsilon _0\) small, and for \(n\) sufficiently high. For the proof of the theorem, it suffices to construct, for each \(\epsilon > 0\), a sequence \(\big \{ \mathrm{C}_n^\epsilon : n \in \mathbb{N }\big \}\) of couplings \(S_p^n\) and \(S_{\alpha + 1/2}\) for which
for then we may find an increasing sequence \(\big \{ n_i: i \in \mathbb{N }\big \}\) such that
for each \(i \in \mathbb{N }\), and set \(\mathrm{C}_m = \mathrm{C}_m^{2^{-k_m}}\), where \(k_m\) is the maximal \(n_i\) that does not exceed \(m\); note that \(k_m \rightarrow \infty \) ensures that this sequence of couplings is as the statement of the theorem demands. A coupling \(\mathrm{C}_n^{\epsilon }\) suitable for (4.15) may be obtained by the use of Proposition 3 to couple the values of \(S_p^n\) and \(S_{\alpha + 1/2}\) at points of the form \(\big \{ i\epsilon T: i = 0,\ldots , \lfloor \epsilon ^{-1} \rfloor + 1 \big \}\), so that the maximum difference \(\vert S_p^n - S_{\alpha +1/2} \vert \) at such points tends to zero in probability under \(\mathrm{C}_n^{\epsilon }\). We then use Lemma 4.2, and the remark following its statement, as well as \(S_{\alpha + 1/2}\) being uniformly continuous on \([0,T]\) to verify (4.15).
\(\square \)
5 Open problems
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It would be interesting to find the negatively correlated fractional Brownian motions, with Hurst parameter \(H\in (0,{1/2})\), as scaling limits of variants of the discrete processes that we consider. A natural first guess is the walk arising from the model in which each vertex of \(\mathbb{Z }\) is given the opposite sign of its parent in \(G_\mu \), instead of the same sign. However, it is quite possible that this gives rise to Brownian motion as a scaling limit of the associated walk. A more promising candidate discrete model is one in which the vertices in each component of \(G_\mu \)—ordered with respect to the natural ordering of \(\mathbb{Z }\)—are given values that alternate between \(1\) and \(-1\), independently choosing one of the two ways of doing this, each with probability \(1/2\).
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As mentioned in the discussion following the statement of Theorem 1.1, we believe that the theorem is sharp. We pose the problem to show that if a measure \(\mu \) on \(\mathbb{N }\) is such that the conclusion of Theorem 1.1 holds, with some deterministic function of \(n\) playing the role of \(\tilde{c} n^{-1/2 - \alpha }L(n)\), then \(\mu \in \Gamma _\alpha \). An important step here would be to show that, if the measure \(\mu \) is such that the variance of \(S_n\) is a slowly varying multiple of \(n^{2\alpha + 1}\), then \(\mu \in \Gamma _\alpha \).
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Define \(\tilde{\Gamma }_{\alpha }\) according to Definition 1.1, with the alteration that \(\mu \) is supported on \(\mathbb{Z }\) and is symmetric about \(0\), and let \(\mu \in \tilde{\Gamma }_{\alpha }\) for some \(\alpha \in (0,\infty )\). Consider a voter model in dimension one, in which voters reside at the elements of \(\mathbb{Z }\). Each voter at a given time has an affiliation to one of two political parties. At any given moment \(t \in \mathbb{Z }\) of time, each voter selects the resident at a displacement given by an independent sample of \(\mu \), and inherits the affiliation that this resident held at time \(t-1\). (A continuous time variant, where opinions are imposed on a voter at a Poisson point process of times, may also be considered). We propose the problem of considering the set of equilibrium measures of this process as an analogue of the family \(\big \{\lambda _p : p \in [0,1] \big \}\) and seeking a counterpart to Theorem 1.1. The two unanimous configurations are always equilibrium measures, and we anticipate that mixtures of these are the only such measures exactly when \(\alpha \ge 1\) (and the condition \(\mathrm{g.c.d.}(\mu ) = 1\) is satisfied). This is because the transmission histories of the affiliation held by two voters will almost surely coincide in the distant past precisely when a discrete time random walk with step distribution \(\mu \) almost surely reaches zero. The case \(\alpha = 1\) corresponds to the step distribution of the \(x\)-displacement of a two-dimensional simple random walk between its successive visits to the \(x\)-axis; such a walk is recurrent, but only “marginally so”, pointing to the value of \(\alpha = 1\) being critical for the problem.
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Let \(d \ge 2\). For any measure \(\mu \) whose support is contained in the integer lattice \(\mathbb{Z }^d\), the random \(\mathbb{Z }^d\)-spanning graph structure \(G_{\mu }\) may be defined. Consider a law \(\mu \) that has a regularly decaying heavy tail. For example, we might insist that \(\mu \big (B_n^c \big ) = n^{-\alpha } L\big (\vert \vert n \vert \vert \big )\), where \(L:(0,\infty ) \rightarrow (0,\infty )\) is slowly varying. Here, \(B_n = \big \{ x \in \mathbb{Z }^d: \vert \vert x \vert \vert \le n \big \}\). If \(\mu \) is chosen to be “symmetric”, in the sense that \(\mu \{ x \} = \mu \{ y \}\) whenever \(x = -y\), it is not hard to show that all the components of \(G_{\mu }\) are finite, each containing a unique cycle. Choices of \(\mu \) supported in a half-plane \(\big \{ x_1 < 0\big \}\) will give rise to infinite components, however. This raises the question of finding a phase transition in \(\alpha \) for uniqueness of the infinite component for some family of measures \(\mu \) supported in the strict half-plane \(\big \{ x_1 < 0\big \}\). In fact, the discussion in the preceding problem already addresses this problem in a certain guise. To obtain the voter model problem, we take \(d = 2\), write points in the plane in the form \((t,x)\), and take \(\mu \) to be supported on the line \(\big \{ x = -1 \big \}\), with \(\mu \{ -1, \cdot \} \in \tilde{\Gamma }_{\alpha }\). Naturally, we might take \(d > 2\). In this case, in seeking a result analogous to Theorem 1.1, we would seek to show that, for regularly varying laws \(\mu \) whose decay is slow enough to ensure that the transition histories of distinct voters may be disjoint, the affiliation “white noise” in the time-slice \(\big \{ t = \mathrm{constant} \big \}\) at equilibrium is given by a fractional Gaussian field.
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Theorem 1.1 states that fractional simple random walk and fractional Brownian motion may be coupled to be close in \(L^{\infty }\) on compact sets under rescaling of the discrete walk. One might seek to quantify this, investigating how quickly \(\epsilon \) may be taken to \(0\) in the limit of high \(n\) in (1.4), for some coupling \(C_n\). The analogue for simple random walk and its convergence to Brownian motion is the celebrated Komlos-Major-Tusnady theorem [18], one case of which states that simple random walk may be coupled with asymptotic probability one to a Brownian motion with a uniform error in the first \(n\) steps of at most a large constant multiple of \(\log n\). Quantifying the rate of convergence could be useful if we would like to compare the stochastic difference equation driven by the fractional discrete process and its continuous analogue driven by fractional Brownian motion, the latter having been extensively studied. (See [5, 9, 20] for treatments of the stochastic calculus of fractional Brownian motion.)
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Acknowledgments
We thank Yuval Peres for suggesting that the walks in question may converge to fractional Brownian motion. We thank S.R.S. Varadhan for discussions relating to regular variation. The research was undertaken while the authors were affiliated to the Courant Institute of Mathematical Sciences at New York University. A. Hammond was supported in part by the US National Science Foundation under grants OISE-07-30136 and DMS-0806180. S. Sheffield was partially supported by NSF grants DMS-0403182, DMS-0645585 and OISE-07-30136.
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Hammond, A., Sheffield, S. Power law Pólya’s urn and fractional Brownian motion. Probab. Theory Relat. Fields 157, 691–719 (2013). https://doi.org/10.1007/s00440-012-0468-6
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DOI: https://doi.org/10.1007/s00440-012-0468-6
Mathematics Subject Classification
- 60G22