Random Gaussian sums on trees

Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=\sigma(t)\sum_{v \preceq t}\alpha(v)\xi_v $$ for given weight functions $\alpha$ and $\sigma$ on $T$ and with $(\xi_v)_{v\in T}$ i.i.d. standard normal. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined via $\alpha$ and $\sigma$, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we investigate the boundedness of $X$ for the binary tree and for homogeneous weights. Assuming some mild regularity assumptions about $\alpha$ we completely characterize weights $\alpha$ and $\sigma$ with $X$ being a.s. bounded.


Introduction
Let T be a finite or infinite tree with root 0 and let " " be the induced partial order generated by the structure of T , i.e., it holds t s or, equivalently s t, whenever t is situated on the branch connecting 0 with s. Suppose we are given two weight functions α and σ mapping T into [0, ∞) with σ non-increasing, that is, σ(t) ≥ σ(s) whenever t s. If (ξ v ) v∈T denotes a family of independent standard normal random variables, then the centered Gaussian process X = (X t ) t∈T with is well defined. Its covariance function R X is given by Fernique was probably the first to consider such summation schemes on trees in his constructions of majorizing measures [3]. More recently, they were extensively studied and applied in relation to various topics, see e.g. the literature on Derrida random energy model [1] or displacements in random branching walks [10], to mention just a few. Moreover, summation operators related to those processes have been recently investigated in [7], [8] and in [9]. Some of the ideas used there turned out to be useful as well for the study of Gaussian summation schemes on trees.
The basic question investigated in this paper is as follows: Given a tree T characterize weights α and σ such that X is a.s. bounded, i.e., that P sup t∈T |X t | < ∞ = 1 . (1.2) In a first part we give necessary and sufficient conditions for the weights in order that (1.2) holds. These results are valid for arbitrary trees and they are based on covering properties of T by ε-balls with respect to a certain metric d first introduced in [8]  It is worthwhile to mention that neither (1.4) nor (1.5) are direct consequences of the wellknown conditions due to R.M. Dudley and V.N. Sudakov (cf. [2] and [11]), respectively. The latter results are based on compactness properties of (T, d X ) with so-called Dudley metric d X defined by d X (t, s) := E |X t − X s | 2 1/2 , t, s ∈ T , (1. 6) and not on d as introduced in (1.3). We shall see below that, in general, the covering numbers w.r.t. d and to d X may behave quite differently. The main advantage of Theorem 1.1 is that in many cases the covering numbers w.r.t. d are easier to handle than those defined by d X (cf. [8] for concrete estimates of N (T, d, ε) and also Corollary 3.4 below).
It is well-known that in general entropy estimates are too rough for deciding whether or not a given Gaussian process is bounded. Only majorizing measure techniques would work. But, unfortunately, majorizing measures are difficult to handle and we do not see how their use leads to a characterization of weights α and σ for which X is a.s. bounded. Therefore, in a second part, we investigate special trees and weights where a direct approach is possible. We suppose that T is a binary tree and that the weights are homogeneous, i.e. α(t) and σ(t) only depend on the order |t| (cf. (2.1) for the definition) of t ∈ T . Our results (cf. Theorems 6.1 and 6.3 below) imply the following: Theorem 1.2 Let T be a binary tree and suppose α(t) = α |t| and σ(t) = σ |t| for two sequences (α k ) k≥0 and (σ k ) k≥0 of positive numbers with σ k non-increasing.
The organization of this paper is a s follows. After a short introduction to trees, Section 3 is devoted to the proof of Theorem 1.1. In Section 4 we thoroughly investigate the relation between the two metrics d and d X . Here the main observation is that N (T, d, ε) and N (T, d X , ε) may behave quite differently. Nevertheless, in view of Theorem 1.1 and the well-known results due to R.M. Dudley and to V.N. Sudakov, on the logarithmic level the covering numbers of these two metrics should be of similar order. We investigate this question in Section 5 more thoroughly.
In particular, we show that ε 2 log N (T, d, ε) is bounded if and only if ε 2 log N (T, d X , ε) is so. In Section 6 we treat processes X indexed by a binary tree and with homogeneous weights. We prove slightly more general results than stated in Theorem 1.2. Finally, we give some interesting examples of bounded as well as unbounded processes indexed by a binary tree. In particular, these examples show that the boundedness of X may not be described by properties of the product α σ only.

Trees
Let us recall some basic notations related to trees which will be used later on. In the sequel T always denotes a finite or an infinite tree. We suppose that T has a unique root which we denote by 0 and that each element t ∈ T has a finite number of offsprings. Thereby we do not exclude that some elements do not possess any offspring, i.e., the progeny of some elements may "die out". The tree structure leads in natural way to a partial order ,, " by letting t s, respectively s t, provided there are t = t 0 , t 1 , . . . , t m = s in T such that for 1 ≤ j ≤ m the element t j is an offspring of t j−1 . The strict inequalities have the same meaning with the additional assumption t = s. Two elements t, s ∈ T are said to be comparable provided that either t s or s t.
For t, s ∈ T with t s the order interval [t, s] is defined by and in a similar way we construct (t, s] or (t, s) . A subset B ⊆ T is said to be a branch provided that all elements in B are comparable and, moreover, if t v s with t, s ∈ B, then this implies v ∈ B as well. Of course, finite branches are of the form [t, s] for suitable t s.
For any s ∈ T its order |s| ≥ 0 is defined by Let ρ be an arbitrary metric on the tree T . Given ε > 0 a set O ⊆ T is said to be an ε-order net w.r.t. ρ provided that for each s ∈ T there is an t ∈ O with t s and ρ(t, s) < ε. Let be the corresponding order covering numbers. Clearly, we have N (T, ρ, ε) ≤Ñ (T, ρ, ε) . 3 Proof of Theorem 1.1 Let T be an arbitrary tree and let α and σ be weights on T as before. Define X = (X t ) t∈T as in (1.1). Of course, whenever (1.2) holds, then we necessarily have Thus let us assume that (3.1) is always satisfied.
In order to prove part one of Theorem 1.1 in a first step we replace the process X by a processX which is easier to handle.
To this end, if k ∈ Z, define I k ⊆ T by and a new weightσ byσ := LetX be the process defined by α andσ via (1.1), i.e., it holdŝ and letd denote the distance generated via α andσ as in (1.3). Then the following are valid.

The process X is a.s. bounded if and only ifX is a.s. bounded.
Proof: The first assertion follows easily by the definition of d andd while the second one is a direct consequence of As a consequence of the preceding proposition we conclude that it suffices to prove Theorem 1.1 in the case of non-increasing weights σ of the form The property that σ is non-decreasing reflects in the following properties of the partition (I k ) k∈Z of T .
1. Whenever B ⊆ T is a branch, then for each k ∈ Z either B ∩ I k = ∅ or it is an order interval in T .
2. If l < k, t ∈ B ∩ I l , s ∈ B ∩ I k , then this implies t ≺ s.
3. I k = ∅ whenever k ≤ k 0 for a certain k 0 ∈ Z.
Thus from now on we may suppose that the weight σ is as in (3.5) with a partition (I k ) k∈Z of T possessing properties (1), (2) and (3) stated before.
In a second step of the proof of Theorem 1.1, first part, we define a process Y := (Y t ) t∈T which may be viewed as a localization of X. To this end let us write t ≡ s provided there is a k ∈ Z such that t, s ∈ I k . With this notation we set It is an easy deal to relate the boundedness of X with that of Y .

Proposition 3.2 The process Y is a.s. bounded if and only if X is a.s. bounded.
Proof: Actually, we establish simple linear relations between Y and X, see (3.7) and (3.8) below. For any integers ℓ ≤ k and any t ∈ I k set B ℓ (t) := [0, t] ∩ I ℓ . Then we have where the last sum is taken over ℓ ≤ k such that B ℓ (t) = ∅ and λ ℓ (t) := max{s : s ∈ B ℓ (t)}. It follows from (3.7) that the boundedness of Y yields that of X.
To prove the converse statement of Proposition 3.2, take an arbitrary t ∈ T and consider two different cases. If t ≡ 0 (recall that 0 denotes the root of T ), then by the definition of Y we simply have Y t = X t .
By the definition of Y t we obtain Since the weight σ is non-increasing, by λ − (t) t we get σ(t) σ(λ − (t)) ≤ 1. It follows from (3.8) that if X is a.s. bounded this is also valid for Y as claimed. This completes the proof.
In the next step we calculate the Dudley distance generated by Y and compare N (T, d Y , ε) withÑ (T, d, ε). Recall thatÑ (T, d Y , ε) andÑ (T, d, ε) are the corresponding order covering numbers as introduced in (2.2).

s. Then it follows that
Proof: If t s, then we get Of course, this implies the second inequality in (3.9), the first one being trivial. Thus the proof is complete.
Proof of Theorem 1.1, first part: Without loosing generality we may assume α(0) = 0. Indeed, write and observe that sup t∈T σ(t) < ∞. Moreover, the metric d is independent of α(0). Note that this number never appears in the evaluation of d(t, s) for arbitrary t, s ∈ T . Thus let us assume now that (1.4) is valid.
Consequently, Dudley's theorem (cf. [2] or [6], p.179) applies for Y and d Y , hence Y possesses a.s. bounded paths. In view of Proposition 3.2 the paths of X are also a.s. bounded and this completes the proof of the first part of Theorem 1.1.
Proof of Theorem 1.1, second part: Take ε > 0. As proved in [8,Proposition 5.2] there are at least m : Next, set Then it follows that and, consequently, the η i are independent centered Gaussian with Since σ is assumed to be non-increasing, we get Suppose now that X is a.s. bounded. By Fernique's theorem (cf. [4] or [6], p.142) this implies and by the choice of m the assertion follows by where we used (3.11) and the classical Fernique-Sudakov bound recalled below in (6.4).
The main advantage of Theorem 1.1 is that there are quite general techniques to get precise estimates for N (T, d, ε) (cf. [8]). For example, Theorem 1.1 implies the following.

Corollary 3.4 Let T be a binary tree and suppose that
for some γ < 1 and σ(t) ≡ 1, then the generated process X is a.s. unbounded.
Remark: The second part of Corollary 3.4 does no longer hold for non-constant weights σ. In different words, an estimate α(t)σ(t) ≥ c |t| −γ with 1/2 < γ < 1 does not always imply that X is unbounded (cf. the remark after Corollary 6.4 below).
Corollary 3.4 suggests that the boundedness of a process with weights α(t) and σ(t) might be determined by the product σ(t)α(t). In other words, it is natural to ask what is the relation between the boundedness of this process and the process generated by the weights σ(t) :≡ 1 and α(t) := σ(t)α(t). It turns out that (only) a one-sided implication is valid.
To investigate this question, for a moment write X α,σ for the process defined in (1.1).
Proposition 3.5 If X ασ,1 is a.s. bounded, then this is also true for X α,σ .
Proof: We only give a sketch of the proof.
1. Recall a general fact from the theory of Gaussian processes: If X and Y are two independent centered Gaussian processes, then X + Y bounded yields X bounded. This is an immediate consequence of Anderson's inequality (cf. [6, p.135]).
2. By applying this fact we obtain: If two weights are related by α 1 ≤ c α 2 and X α 2 ,1 is bounded, then X α 1 ,1 is bounded as well.
4. Let now Y be the process constructed in the article associated to X α,σ .
Recall that we know that the boundedness of Y is equivalent to that of X α,σ .
Examples in Section 6 show that the statement of Proposition 3.5 cannot be reversed, i.e., in general the boundedness of X α,σ does not yield that of X ασ,1 .
4 Compactness properties of (T, d) versus those of (T, d X ) The aim of this section is to compare the metric d on T defined in (1.3) and the Dudley distance d X introduced in (1.6). For X defined by (1.1) the latter distance equals In particular, if t s, this reads as Recall, that for t s we have Comparing (4.1) with (4.2), it is not clear at all how these two distances are related in general.
In a first result we show that the covering numbers w.r.t. d and to d X may be of quite different order. Proposition 4.1 There are non-increasing weights α and σ on a tree T such that the generated process X is a.s. bounded and, moreover, Proof: Take T = N 0 = {0, 1, . . .} and let α(0) = σ(0) = 1. If k ≥ 1 set α(k) = k −ν and σ(k) = k −θ for some θ, ν > 0, i.e., The law of iterated logarithm tells us that the process X is a.s. bounded if and only if θ+ν > 1/2. Thus let us assume that this is satisfied. Take now any 1 ≤ k < l. Then by (4.1) it follows Hence, if 1 ≤ k < l ≤ n for some n ≥ 2, this implies for some c > 0 only depending on θ.
Comparing (4.9) with (4.4) shows that for ν > 1 estimate (4.8) cannot lead to sharp estimates while, as seen above, the use of N (T, d, ε) does so. In some sense this observation proves that the metric d fits better to those processes X than d X does.
In spite of this observation we will show now that d X (t, s) may become arbitrarily small while d(t, s) ≥ C > 0. Hence an estimate d(t, s) ≤ c d X (t, s) cannot be valid in general. Recall that in view of Proposition 4.1 a relation d X (t, s) ≤ c d(t, s) is impossible as well.
Proposition 4.2 There are weights α and σ on T = N 0 such that the corresponding process X is a.s. bounded and such that lim k→∞ d X (0, k) = 0 while d(0, k) = C > 0 for all k ≥ 1.

More about the relation between processes and their entropy
So far, we came out with a somewhat messy set of relations between the processes and their entropy. Let us try to rearrange it again and to put the things in order. We have three consecutively generated processes X →X → Y . In this section, we will not identify or replace X byX, as we did sometimes before.
Before proceeding further, let us make a useful and well-known identification. Suppose X = (X t ) t∈T is an arbitrary Gaussian process (in fact we only need that it is a process with finite second moments) modeled over a probability space (Ω, A, P). Then we may regard X as subset of the Hilbert space L 2 (Ω, A, P), i.e., we identify X with {X t : t ∈ T } and the induced distance equals In particular, we may also build the absolutely convex hull of X in L 2 (Ω, A, P) which we denote by aco(X).
Suppose now that the processes X,X and Y are defined as in (1.1), (3.4) and (3.6), respectively, where for the construction ofX and Y we use the partition (I k ) k∈Z given by (3.2). First we show that aco(X), aco(X), and aco(Y ) are the same sets up to a numeric constant. Namely, the following is valid. Remark: Clearly, (5.1) means that the three processes are either all bounded or all are unbounded. But, certainly, it contains even more information.
Now we move to covering numbers in order to clarify the role of the distance d defined in (1.3). We will show now that on the logarithmic level there is no much difference between N (X, || · || 2 , ε) = N (T, d X , ε) and N (T, d, ε).
Proof: We first give the lower bounds for N (T, d X , ε). By (3.10) and (3.11) it follows that Our next task is to replace X − [0, 1] · X by X in (5.6) by using the following trivial fact.
Proof of the lemma: Let B be an ε-net for X and set Clearly, Take any z = θx ∈ [0, 1] · X with x ∈ X, θ ∈ [0, 1]. Find y ∈ B and a positive integer j ≤ M X ε such that Then z ′ := jε M X y ∈ C and observe that Hence, C is a 2ε-net for [0, 1] · X and the first claim of the lemma is proved. The second one follows immediately.
By combining the preceding estimates we finish the proof.

The binary tree with homogeneous weights
Before investigating Gaussian processes on binary trees let us shortly recall some basic facts about suprema of Gaussian sequences. Let (X 1 , . . . , X n ) be a centered Gaussian random vector. Introduce the following notations: and let m S be a median of S. Then the following is well known.
• The following concentration principle is valid: is the standard Gaussian tail. See [6], p.142. By combining this with (6.1) we also have • It is true that ES ≤ 2 log n σ 2 .
• If X 1 , . . . , X n are independent, then ES ≥ c log n σ 1 . Remark that the same properties hold true for Let T be a binary tree and suppose that the weights depend only on the level numbers, i.e. α(t) = α |t| and σ(t) = σ |t| for some sequences (α k ) k≥0 and (σ k ) k≥0 of positive numbers with (σ k ) k≥0 non-increasing. The following two theorems give, with a certain overlap, necessary and sufficient conditions for the boundedness of (X t ) t∈T in that case. Proof: a) Let us construct a random sequence (t n ) n≥0 in T and a sequence of random variables (ζ n ) n≥1 by the following inductive procedure. Let t 0 = 0. Next, assuming that t n is constructed, let t ′ and t ′′ be the two offsprings of t n . We let It is obvious that (ζ n ) are i.i.d. random variables with strictly positive expectation. Our construction yields where C := Eζ j > 0. Since the assumption "(X t ) t∈T is a.s. bounded" implies E sup t∈T X t < ∞, we obtain (6.5).
b) Let us assume that G < ∞, Q < ∞ and prove that (X t ) t∈T is a.s. bounded.
where U m := sup By using that (σ k ) k≥0 is non-increasing, we infer from (6.7) for any M ≥ 0 and t ∈ J M We will use now standard Gaussian techniques in order to evaluate the quantities on the r.h.s. Note that on the binary tree Assuming (6.6) to hold, we obtain Using (6.6) again we arrive at Now by (6.3) it follows that Hence, for any M we get On the other hand, by the Gaussian concentration principle (6.2), From (6.9) it follows that By plugging this into (6.8), we arrive at and (X t ) t∈T is a.s. bounded.

Corollary 6.2 The process
is a.s. bounded while is a.s. unbounded.
Proof: In the first case (6.5) and (6.6) are satisfied while in the second one (6.5) fails. Thus both assertions follow by Theorem 6.1.

Remark:
The preceding corollary is of special interest because α(t)σ(t) = (|t| + 1) −1 in both cases. Consequently, the boundedness of the process X cannot be described by the behavior of ασ. This is in contrast to the main results about metric entropy in [8] which only depend on this product behavior.
Theorem 6.1 does not apply in the case of rapidly increasing sequences (α k ) k≥0 because (6.6) fails for them. The next theorem fills this gap.  Notice that the (Y t ) |t|=m are independent and that We arrive at , and achieve the proof of a) by taking the supremum over m and n.
b) Let S n := max |t|=n X t . By (6.3) we have We also have Since sup t∈T X t = sup n S n , for any r > 0 it follows that P (S n ≥ ES n + r) (by (6.12) ) (by (6.13) and (6.2) ) It follows that (X t ) t∈T is a.s. bounded. Thus assertion b) is proved.
c) The inequality G 1 ≤ G 2 is obvious for any (α k ) k≥0 . We only need to show that a bound in the opposite direction holds, too. Let : n odd.
Assuming that (α k ) k≥0 is non-decreasing, we have It follows that Corollary 6.4 Let α k = k b 2 k for some b ∈ R. Then (X t ) t∈T is a.s. bounded if and only if sup n σ n n 1/2+b 2 n < ∞.
Remark: Note that criterion (6.5) from Theorem 6.1 fails to work in that case. Moreover, letting b = −γ with 1/2 < γ < 1 and σ n = 2 −n , by Corollary 6.4 the corresponding process is bounded although α(t)σ(t) ≥ |t| −γ for t ∈ T . This shows that the second part of Corollary 3.4 is no longer valid for non-constant weights σ.
Another example where Theorem 6.1 does not apply is as follows. An application of Theorem 6.3 yields the result.
Our message is that Theorems 6.1 and 6.3 should jointly cover any reasonable case. Let us illustrate this by the following example. Recall that by the first part of Corollary 3.4, if T is the binary tree and α(t)σ(t) ≤ c |t| −γ for some γ > 1, then the generated process X is a.s. bounded. For homogeneous (level-dependent) weights this means that α k σ k ≤ c k −γ for some γ > 1 yields the a.s. boundedness of X. Let us see how this fact is related to Theorems 6.1 and 6.3.
Essentially, we have the following • If (α k ) k≥0 is decreasing, then hence (6.5) and (6.6) hold and Theorem 6.1 yields the boundedness.
Finally let us relate the results in Theorems 6.1 and 6.3 to those about compactness properties of (T, d) with d defined in (1.3). Here we have the following partial result. for all 0 ≤ m < n < ∞. Suppose now that (6.14) holds and take integers m < n. Next choose two elements t, s ∈ T with t ≺ s such that m = |t| and n = |s|. Note that (6.14) implies σ n n k=m+1 α 2 k 1/2 ≤ d(t, s) ≤ c |t| −1/2 = c m −1/2 which proves (6.15).
Conversely, assume (6.15) and take any two elements t ≺ s in T . Furthermore, let v ∈ (t, s] be a node where Remark: Clearly (6.14) implies log N (T, d, ε) ≤ c ε −2 as we already know by combining Theorems 1.1 and 6.3. But it says a little bit more. Namely, an ε-net giving this order may be chosen as t ∈ T : |t| ≤ c ε −1/2 for a certain c > 0. Of course, this heavily depends on the fact that we deal with homogeneous weights.