Small deviations in lognormal Mandelbrot cascades

We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if $\log \log 1/P(W \leq x) \sim \gamma \log \log 1/x$ as $x \to 0$ with $\gamma>1$, then the Laplace transform of $Y$ satisfies $\log \log 1/\E e^{-t Y} \sim \gamma \log \log t$ as $t \to \infty$. As an application, this gives new estimates for $\Prob(Y \leq x)$ for small $x>0$. As another application of our methods, we prove a similar result for a variable arising as a total mass of a lognormal $\star$-scale invariant multiplicative chaos measure.


Introduction
We start by outlining the problem studied in this note. The Mandelbrot cascade on a binary tree is the following construction. Let W be a given positive 1 random variable such that EW = 1/2 and {W σ } σ∈Σ be an i.i.d. collection of copies of W indexed by the infinite binary tree Σ = n≥1 {0, 1} n . The n-th cascade variable 2 Y n is defined by Y n = σ 1 σ 2 ...σn∈{0,1} n W σ 1 W σ 1 σ 2 . . . W σ 1 σ 2 ...σn .
The sequence (Y n ) n≥1 is a positive martingale and as such almost surely convergent to a limit variable Y = lim n→∞ Y n which satisfies the functional equation where the variables W 0 , W 1 , Y 0 and Y 1 are independent and W 0 (1) holds is called a fixed point of the smoothing transform associated to W . For a given positive random variable W , the fixed points of the smoothing transform have been characterized by Durrett and Liggett [7]. The specifics of the characterization do not concern our study here, but the following facts are good to Date: May 7, 2014. 1 By positive we mean P(W > 0) = 1. 2 To all n ∈ N one may also associate a random measure Mn on [0, 1] by giving the dyadic interval naturally encoded by σ ∈ {0, 1} n the mass Wσ 1 Wσ 1 σ 2 . . . Wσ 1 σ 2 ...σn . However, we will not work with these cascade measures and mention them only for motivation. know. First, if the smoothing transform given by W has fixed points of finite mean, all the fixed points are given by constant multiples of the Mandelbrot cascade associated to W , and in the case of infinite mean the Mandelbrot cascade may (under rather general assumptions) be deterministically renormalized in order to obtain the fixed points [2,20,13]. Second, for a positive W the fixed points of the smoothing transform are also positive, i.e. have P(Y > 0).
The study of the tail of the fixed points of smoothing transforms at positive infinity has a long history. Indeed, the finiteness of the moments EY p , p > 1, was a central question already in the work of Mandelbrot [15] on his cascades, answered by Kahane and Peyrière [12]. The behavior of the Laplace transform Ee −tY near 0 was used by Durrett and Liggett in the characterization of the fixed points of the smoothing transforms.
Given that P(W > 0) = 1 = P(Y > 0), the asymptotics of the probabilities of Y being small are also of interest. To mention just some work on this question, in connection with the multifractal analysis of Mandelbrot cascade measures it was shown by Holley and Waymire [8] that if there exists an a > 0 such that P(W ≥ a) = 1, the Laplace transform of cascade variable Y satisfies Ee −tY ≤ exp −ct b for some constants c > 0 and 0 < b < 1 depending on the distribution of W . In a more general study of multifractal analysis of Mandelbrot cascade measures, Molchan [16] proved that if EW −q < ∞ for some q > 0, then also EY −2q < ∞. These results have been later improved by Liu [10] and most recently Hu [9], who have shown much stronger results relating the asymptotics of P(W ≤ x) near 0 to the asymptotics of Ee −tY near ∞ in the more general case of a smoothing transform in which the number of summands W i Y i appearing on the right hand side of (1) is random and the i.i.d. assumption on the (W i ) is relaxed.
However, there is little earlier work on the asymptotics of Ee −tY near ∞ in the important special case where the generator W is lognormal: the best result in the literature seems to be Molchan's result on finiteness of the moments of negative order. The lognormal generator was considered already in the original work of Mandelbrot [14], but interest in specifically lognormal cascades has been revitalized recently by the analogies between lognormal Mandelbrot cascades and lognormal multiplicative chaos, a much more general construction of a random measure given originally by Kahane [11]. Kahane's lognormal multiplicative chaos measures have recently been connected to problems in mathematical physics involving the exponential of the Gaussian free field, Liouville quantum gravity and the KPZ formula; we refer the reader to the recent survey of Rhodes and Vargas [19] for details on these connections.
Finally, we mention the work of Ostrovsky [17] in which a prediction is made for the exact form of the Mellin transform of the law of the total mass of a certain lognormal multiplicative chaos measure. While the accuracy of the asymptotics given by our result is certainly far from providing a rigorous proof to Ostrovsky's formula, it is worth noting that our result is in accordance with the prediction.
Our main result is Theorem 1 below, which connects the asymptotics of P(W ≤ x) as x → 0 for a class of random variables (that includes the lognormal variables) to the asymptotics of Ee −tY as t → ∞. This theorem is then applied to fixed points of the smoothing transform (Theorem 2) and thus to (possibly renormalized) Mandelbrot cascades, and then to a simple example of multiplicative chaos measures (Theorem 7).
The notation X Z means that X is stochastically dominated by Z, i.e. that P(X ≥ x) ≤ P(Z ≥ x) for all x ∈ R. Theorem 1. Let W and Y be positive random variables satisfying is an independent pair of copies of Y , independent of (W 0 , W 1 ), Suppose further that there exist γ > 1 and x ′ ∈]0, 1[ such that Then for any α ∈ [1, γ[ there exists a constant t α > 0 such that for all t ≥ t α we have Remarks. Note that W 0 and W 1 are not assumed to be independent. The assumption W 0 d = W 1 could easily be relaxed, in which case instead of the assumption (3) one would assume that the lighter of the negative tails of log W 0 and log W 1 would satisfy a similar condition. Similarly, instead of two summands W i Y i , i = 0, 1, we could consider an arbitrary but fixed finite number N of summands.
Theorem 2. Suppose W is a positive random variable satisfying Let Y be the fixed point of the smoothing transform associated to W or in other words the limit variable of the (possibly renormalized) Mandelbrot cascade generated by W , i.e. suppose that Then We state the resulting estimate for P(Y ≤ x) as a corollary.
Corollary 3. Let W and Y be as in Theorem 2. Then also In Section 3 we exhibit an example of a lognormal multiplicative chaos measure for which we state and prove a corresponding result.

Proof of Theorem 1
The proof of Theorem 1 proceeds by starting from the result of Molchan on the finiteness of moments of Y of negative order and using this information to get better bounds on the decay of the Laplace transform of Y near infinity. This procedure is then iterated, giving better and better estimates. We state the following proposition, which adapts the methods of Liu [10] and Barral [3] to this iteration procedure, as a result of its own. Proposition 4. Suppose W and Y satisfy (2) and (3) and further that for Proof. Throughout the proof we will for brevity denote φ(t) = Ee −tY and for all t ≥ t α . It follows that there exist constantst α > 0 and C α > 0 such that The main estimate of the proof is derived next. We use the distributional inequality (2) and the Cauchy-Schwarz inequality to deduce It follows that for all t ≥ t ′ > 0 we have Using this observation witht = tW andt ′ = t 1/2 ≥ 1, we get the estimate where W 1 and W 2 denote independent copies of W . We plug this estimate into (6) to obtain, for all t ≥ 1, Moreover, by using (6) as above, for an arbitrary positive random variable where W ′′ d = W is independent of V and W ′ . Let (W n ) n≥1 be an i.i.d. sequence of copies of W and define the stopping time . . we obtain, for any n ∈ N and all t ≥ 1, It remains to compute that the estimate (9) indeed gives (4) for α ′ ∈ [α, α + (γ − α)/(γ + 1)]. For any t ≥ 1, the probability P(τ t = k) may be estimated by The decay rate (5) derived for ψ in turn gives, for t ≥t 2 α , and therefore the terms of the sum in (9) may be estimated by for t ≥t 2 α and k ∈ N satisfying (10). Finding the minimum of k → f t (k) on ]0, ∞[ is easy: the zero of the derivative is at where the constants C ′ , C ′′ > 0 and C = C ′ + C ′′ only depend on the constants γ, c, α and C α . Plugging this into (11) gives for t ≥t 2 α and k ∈ N satisfying (10). Next we estimate the final term in (9) and choose the value of n. It is enough to use the crude estimates φ ≤ 1 and P(τ t > n) ≤ 1 and the decay rate (5) to get for t ≥t 2 α . Choosing n = (log t) γ−α γ+1 in (9), we see that (10) is satisfied for all sufficiently large t and k ≤ n, so by (12) and (13) we have shown that there exists at ≥t 2 such that for all t ≥t .
For any α ′ ∈ [α, α + (γ − α)/(γ + 1)], the prefactor in the first term above may be absorbed in order to obtain the desired constants c α ′ , t α ′ > 0 for which we have the estimate The proof is complete.
Theorem 1 now follows by iteration.
Proposition 5. Suppose the positive random variable W satisfies, for some γ > 1 and x ′ ∈]0, 1[, (14) P and that Y satisfies where (W 0 , W 1 ) and (Y 0 , Y 1 ) are independent pairs of copies of W and Y , independent of each other. Then there exist constants t γ , c γ > 0 such that Proof. By (15), for all t ≥ 1 we have Iterating this estimate, for all n ∈ N we have Let n be the greatest integer such that t −2 −n ≤ x ′ , i.e. the unique integer such that With this choice (17) gives Since γ > 1, it is now clear that there exist constants t γ , c γ > 0 such that

Applications to Mandelbrot cascades and lognormal multiplicative chaos
We present applications of the preceding analysis to two cases of interest. Theorem 2, the first application, concerns fixed points of the smoothing transform and thus applies to Mandelbrot cascades.
Proof of Theorem 2. Let W be a positive random variable satisfying For any γ − , γ + such that 1 < γ − < γ < γ + there exists a x ′ ∈]0, 1[ for which From Theorem 1 and Proposition 5 it follows that there exist constants t ′ , c γ − , c γ + > 0 such that It follows that Since γ − < γ and γ + > γ are arbitrary, this implies the claim.
Our estimate for the decay of the Laplace transform Ee −tY as t → ∞ results in an estimate for the probabilities P(Y ≤ x) as x → 0, as stated in Corollary 3.
Proof of Corollary 3. An sufficient upper bound for P(Y ≤ x) is given by Markov's inequality: For the lower bound we use the estimate By Theorem 2, for any γ + > γ, for all x > 0 small enough we have for all x > 0 small enough. Thus log log 1/P(Y ≤ x) ≤ log (log 2 + 2 γ + (log 1/x) γ + ) , which implies Together the bounds (18) and (19) imply the claim.
We then consider an application of Theorem 1 to lognormal ⋆-scale invariant multiplicative chaos. We refer the reader to the recent survey [19] of Rhodes and Vargas for an introduction to this class of random measures, and give here an application to a particular random measure on R that is both simple enough to have a compact definition, yet which illustrates the range of applicability of Theorem 1.
Definition 6. Let M be a positive random measure on R d that satisfies the distributional scaling relation where M ε is a positive random measure on R d with the law given by and (ω ε (x)) x∈R d ,ε∈]0,1] is a Gaussian process.
If M satisfies the scaling relation above for a given Gaussian process ω, we say that M is lognormal ⋆-scale invariant.
Under certain conditions on the process ω, it has been shown that a nontrivial lognormal ⋆-scale invariant M can be constructed as lognormal multiplicative chaos, a construction of a positive random measure given by Kahane in [11] and recently extended to the so-called critical case by Duplantier, Rhodes, Sheffield and Vargas [5,6]. Conversely Rhodes, Sohier and Vargas [18] have shown that Kahane's construction gives (essentially) all the stationary lognormal ⋆-scale invariant random measures such that the masses of open sets have finite moments of order 1 + δ for some δ > 0. Constructions of lognormal ⋆-scale invariant with infinite expectations of masses of open sets have been given, but their (essential) uniqueness has yet to be proven.
We then briefly summarize the construction of the lognormal ⋆-scale invariant random measure to be considered below and refer the reader to [4] for a more detailed exposition for this kind of a construction. We take d = 1 and ω as defined by where β > 0 is a parameter and X is a centered Gaussian process with the covariance The field X may be visualized by considering white noise W on the upper half-plane with control measure dλ = dx dy/y 2 , and integrating W on the truncated triangles i.e. taking X ε (x) = W (T ε (x)). The ⋆-scale invariant random measures associated to ω defined this way may be constructed as and it can be shown that, restricting the measures involved to an arbitrary bounded interval, the limit M exists almost surely in the sense of weak convergence of measures. For β < √ 2 the limit M is almost surely positive on any interval but for β ≥ √ 2 the limit is almost surely null. In the critical case β = √ 2 a ⋆-scale invariant measure is obtained as a limit in probability by renormalizing the density by log 1/ε, see [5,6], and it is expected that for β > √ 2 another deterministic renormalization will in the distributional limit result in a ⋆-scale invariant measure though this is yet to be proven. The theorem below applies to any of these measures (i.e. for any β > 0), but for notational convenience we state it for β ∈]0, √ 2[. The following estimate is derived from Theorem 2 in the same way as Corollary 3 from Theorem 2. Proof of Theorem 7. The required lower bound for the Laplace transform follows from a comparison to a lognormal Mandelbrot cascade using Kahane's convexity inequalities and Proposition 5. This kind of a comparison between multiplicative chaos and Mandelbrot cascades has been utilized already by Kahane [11] and more recently, for example, in [4,5] so we will only sketch the argument. One constructs a Gaussian field (Y ε (x)) x∈[0,1],ε∈]0,1] in such a way that the covariance of Y is dominated by the covariance of X, i.e. EY ε (x)Y ε (y) ≤ EX ε (x)X ε (y) for all 0 ≤ x, y ≤ 1 and 0 < ε ≤ 1, and that the measureM  e ω 1/3 (x) .
The pairs (W 0 , W 1 ) and (Y 0 , Y 1 ) are independent of each other, since ω and M 1/3 are, by ⋆-scale invariance, independent. The stationarity of (X 1/3 (x)) x∈[0, 1] implies that W 0 d = W 1 . It follows from our construction that are independent of each other. It follows that Y 0 and Y 1 are independent. Finally, from ⋆-scale invariance we see that Y 0 To apply Theorem 1, all that remains is to bound P(W 0 ≤ w).
The value of W 0 is determined by the minimum of the centered Gaussian field X 1/3 on [0, 1/3]. Good bounds for the probabilities of extremal values of a Gaussian process being large are given by the Borell-Tsirelson-Ibrahimov-Sudakov inequality (see e.g. [1]), though the following bound can certainly be obtained from weaker results. In our case, since the covariance EX 1/3 (x)X 1/3 (y) is bounded and Lipschitz on (x, y) ∈ [0, 1/3] 2 , there exist constants c ′ , C, A > 0 such that for all a ≥ A we have P inf A short computation shows that this is equivalent to for sufficiently small values of w > 0. For 0 < c < c/β 2 we thus have w ′ > 0 such that P(W 0 ≤ w) ≤ exp −c(− log w) 2 for all 0 < w ≤ w ′ . By Theorem 1, we conclude that for any 1 ≤ α < 2 there exist constants t α , c α > 0 such that Ee −tM ([0,1]) = Ee −tY ≤ exp (−c α (log t) α ) for all t ≥ t α .
The theorem follows from (20) and (21) just as in the proof of Theorem 2.