CLT for Kostlan Shub Smale polynomial systems Joint work with D. - - PowerPoint PPT Presentation
CLT for Kostlan Shub Smale polynomial systems Joint work with D. Armentano; J-M Azas & J. Len Federico Dalmao Random Waves in Oxford 21 June 2018 Plan of the Talk In this talk we are concerned with the asymptotic distribution of the
Joint work with D. Armentano; J-M Azaïs & J. León Federico Dalmao
21 June 2018
In this talk we are concerned with the asymptotic distribution of the geometric measure of the set of real roots of a polynomial system.
◮ Introduction ◮ The model and the results ◮ Dimension 1 ◮ The general case ◮ Chaotic expansion ◮ Bounding the tail
Roots of random polynomials have been intensively studied. However, the case of roots of random polynomial systems have received less attention. Most of the literature is concerned with the Kostlan - Shub - Smale model (and related ones) and to its mean behaviour:
◮ Kostlan, Shub and Smale computed the expectation of
the number of real roots in the square case.
◮ Kostlan and Prior computed the expectation of the
volume of the zero set in the rectangular case.
◮ Burgisser computed the expectation of the intrinsic
volumes (including: number or volume, Euler characteristic, etc).
In the case of the variance:
◮ Wschebor gave asymptotics in the case where the
system’s size m tends to ∞ (r = m).
◮ Letendre (r < m) and Letendre & Puchol and us (r = m)
gave the asymptotics in the case where the common degree d tends to ∞. There are some results for systems In the case of random waves;
◮ D-Nourdin-Peccati-Rossi (arithmetic). ◮ Nourdin-Peccati-Rossi (planar).
For r, m ∈ N with r ≤ m, consider homogeneous polynomials Pℓ(x) =
a(ℓ)
j xj;
ℓ = 1, . . . , r, where we use the notations
◮ j = (j0, . . . , jm) ∈ Nm+1; ◮ x = (x0, . . . , xm) ∈ Rm+1; ◮ |j| = m k=0 jk and xj = m k=0 xjk k . ◮ a(ℓ) j
∈ R.
We assume that the random variables a(ℓ)
j
are independent cen- tered Gaussian with variances Var
j
d j
d! m
k=0 jk!.
The key point is that this choice entails a simple and nice co- variance for Pℓ, namely: rd(s, t) := E(Pℓ(s)Pℓ(t)) = s, td .
Why this model? It has several good properties:
◮ (algebraic) This distribution "is natural" since it is
induced by the Frobenius inner product for matrices.
◮ (geometric) The distribution of the polynomials is
invariant under the action of the orthogonal group in Rm+1.
◮ (probabilistic) There are lot of independences, among the
polynomial and its derivatives, among most of its derivatives.
We are interested in the real roots of the system P = (P1, . . . , Pr). We denote Vd the number of real roots of P in the case r = m
the sphere). As we saw before, there exists 0 < V∞ < ∞ such that lim
d→∞
Var(Vd) dr−m/2 = V∞.
As d → ∞ we have that Vd − E(Vd) d
1 2 (r−m/2)
converges in distribution towards a centered normal random variable with variance V∞.
The one dimensional case P(x) =
d
ajxj; aj independent ∼ N
d j
is simpler. Indeed, counting the number of real roots of P is equivalent to counting the number of roots on the sphere S1 of its homoge- neous version P0 and this is equivalent to counting the number
Xd(t) =
d
aj cosj(t) sind−j(t),
The process Xd is centered, Gaussian and stationary with co- variance function given by rd(t) = cosd(t). In particular, after the scaling x := √ d · t, there exists a limit process (in the weak sense) on [0, ∞); namely a centered, sta- tionary, Gaussian process with covariance given by r∞(x) = exp
2x2
The variance is dealt with using Kac-Rice formula. Wiener Chaos techniques give the convergence of the finite di- mensional projections of the number of real roots of Xd. The tail of the expansion can be bounded by dividing the interval [0, 2π √ d] in small isometric pieces and by approximating the number of real roots of Xd in the pieces by that of the limit process X.
In the general case, working on the sphere Sm complicates things. Nevertheless, Kac-Rice formulae and the Hermite expansion still
Kac-Rice formula: E(V 2
d ) = E(Vd) +
d)(s, t))ps,t(0, 0)dsdt,
being
◮ det ⊥(M) =
◮ Es,t(·) = E(· | Pd(s) = Pd(t) = 0), ◮ ps,t the joint density function of Pd(s) and Pd(t).
Let
◮ γ = (α, β) ∈ Nm × Nm×m; |γ| = i αi + ij βij; ◮
Hγ =
i Hαi · ij Hβij; ◮ cγ is the product of Hermite coefficients of δ0 and det ⊥.
Then, it holds that
d
1 2 (r−m/2)
=
∞
Iq,d, Iq,d =
cγ
The asymptotic variance of Vd is gotten by Kac-Rice formula (J-M’s talk). In order to state the convergence towards a normal random variable of the normalized volume of the zero set of Pd, we take advantage of the expansion and the particular structure of chaotic random variables. (Giovanni’s talk). We study separately the partial sums and the tail
Iq,d and
Iq,d.
The asymptotic normality of the finite partial sums of the ex- pansion is proved using the so called contractions (the Fourth moment theorem by Peccati et al). The moral is that in the case of a sequence of random variables living in a fixed chaos (ie: generated by Hermite polynomials of fixed degree), convergence to the normal distribution is equiva- lent to the convergence of the fourth moment. The contractions provide another technical equivalent mean to prove the normality (let me skip the details).
More precisely, it suffices to prove that: For k = 0, 1, 2, let r (k)
d
indicate the k-th derivative of rd : [−1, 1] → R. If lim
d→+∞ dm/3
π/2 sinm−1(θ) |r (k)
d (cos(θ))| dθ = 0,
then, Iq,d converges in distribution towards a centered normal random variable. Within the sum of a finite number of chaos, joint convergence follows from marginal convergence; so the partial sums con- verges towards a normal rv.
The main issue is the uniform (w.r.t. d) bound of the variance
Actually, the variance of each Iq,d has the form
since it dependes on E( Hγ(s) Hγ(t)) and ps,t(0, 0), which in turns depends on Hq depends on s, td and of its derivatives.
In order to get the global bound, we divide the sphere in the diagonal {(s, t) ∈ Sm × Sm : s = t} and the off-diagonal regions. We first note that outside a
γ √ d -tube of the diagonal the vari-
ance of the tail of the expansion can be bounded directly using Arcones’ inequality. Let X ∼ N(0, Id) on RN and h : RN → R such that E[h2(X)] < ∞. The Hermite rank, rank(h), of h is defined as inf{τ : ∃ k ∈ NN , |k| = τ ; E[(h(X) − Eh(X))Hk(X)] = 0}.
Let W = (W1, . . . , WN) and Q = (Q1, . . . , QN) be two stan- dard Gaussian random vectors on RN. We define r (j,k) = E[WjQk]. Define ψ := max
1≤j≤N N
|r (j,k)|, max
1≤k≤N N
|r (j,k)|
Then, |Cov(h(W), h(Q))| ≤ E[h2(W)] ψrank(h). Using this we bound the tail of the variance by that of a geo- metric series.
It does not exist a global limit process. In fact, denoting by dist the geodesical distance on Sm, we have rd(s, t) = s, td = cos(dist(s, t))d, Using the scaling (s, t) → √ d(u, v), the limit "covariance" is: exp
2
Unfortunatelly, this function does not define a covariance on the sphere Sm.
Let C(s, γ) denote a cap (geodesic ball) C(s, γ) = {t ∈ Sm : dist(s, t) < γ}. After projecting on the tangent space at s, the function exp
2
does define a limit process: the local limit process. This fact allows to bound uniformly the variance of the tail of the expansion of the volume of the zero set on any cap C(s, γ) with small γ. The next step is to transform this local bound into a global one.
Note that, by the invariance of the distribution of the polyno- mials Pℓ, the law of the volume of the zero set restricted to any cap C(s, γ) does not depend on s. We use a convenient partition of the sphere such that, save a negligible set, the sets in the partition, once projected on the tangent space, are asymptotically isometric. The idea is to use hyper-spherical coordinates and to their inter- val of variation taking into account the jacobian of the change
We cover the tube by products of the sets in the partition and bound the variance on each one of them conveniently. More precisely, we prove the convergence of the variance of Vd and of its projections on each chaos to the corresponding ones
This allows us to prove that there exist d0 such that lim
Q→∞ sup d≥d0
Var
Iq,d
Having this in mind, the CLT follows from the Gaussian limit of each partial sum of the expansion of Vd.