18.40.6 Binomial kernels3220889B229A

For a fixed integer $j\geq 0$, set $d=km$, $t_{m,j}=\max \{ \ell _ m-j,0\} $ and $\mathcal H_ m^{(j)}=H^0(B,\mathcal O(d)(-t_{m,j}Z))$. The choices $j=0$ and $j=r$ give $W_ m$ and $V_ m$. Give these spaces the reference norm defined by the Fubini–Study metric, zero continuous weight and the volume $\nu _0=\omega _ B^ q$. For a binomial random variable $\operatorname {Bin}(d,R)$ with $d$ trials and success probability $R$, their kernels are

To see this, apply the monomial norm formula (18.38.3) in degree $d$. Invariance under the coordinate circle actions makes the monomials orthogonal, and the selection keeps exactly those in which $Z_0$ has exponent at least $t_{m,j}$. Summing over the other exponents with the multinomial identity gives (18.40.7). Counting the selected monomials gives $\dim \mathcal H_ m^{(j)}=\binom {d-t_{m,j}+q}{q}=C_ cm^ q+O(m^{q-1})$.

We next bound the tails from below. Write $D(v\mid R)=v\log (v/R)+(1-v)\log ((1-v)/(1-R))$ for the relative entropy, and put $b=\lfloor da\rfloor =\ell _ m$. If $R\geq b/d$, the binomial mode $\min \{ d,\lfloor (d+1)R\rfloor \} $ is at least $b$. Since $t_{m,j}\leq b$, the tail above $t_{m,j}$ contains this mode and is therefore at least $1/(d+1)$. If $0<R<b/d$, the mass at $b$ for the parameter $v=b/d$ is a largest mass, hence at least $1/(d+1)$. For the parameter $R$, the same mass is at least $(d+1)^{-1}\exp (-dD(v\mid R))$. Since

and $v\leq a$, it is at least $(d+1)^{-1}\exp (-dD(a\mid R))$. Together with $u_ c\leq 0$, these cases give $Q_{m,j}\geq (d+1)^{-1}\exp (mu_ c)$, including the cases $b=0$ and $R=1$. Since $A_ d\geq d+1$, the reference kernels satisfy $m^{-1}\log B_{m,j}^0\geq u_ c$; at $R=0$ this holds in the extended sense.

For the matching upper asymptotics, fix $0<R<a$. Stirling's formula at the first selected index $t_{m,j}$ gives the logarithmic rate $-kD(a\mid R)$. Since $t_{m,j}/d\to a>R$, the ratios of successive binomial masses above that index are eventually bounded by a number less than one, so the whole tail has the same logarithmic rate. For $R>a$, the binomial variance estimate gives $Q_{m,j}\to 1$. At $R=a$, the lower bound by the mode and $Q_{m,j}\leq 1$ give the logarithmic limit zero. The functions $m^{-1}\log Q_{m,j}(R)$ increase with $R$, and their limit $u_ c$ is continuous on each interval $[\delta ,1]$ with $\delta >0$. On a finite partition of such an interval on which the oscillation of $u_ c$ is small, monotonicity upgrades pointwise convergence to uniform convergence.

The event that the first $t_{m,j}$ trials succeed gives

Since $\log R$ is integrable against every smooth volume, dominated convergence proves global $L^1$ convergence of the functions $m^{-1}\log Q_{m,j}$, and the additional term $m^{-1}\log A_ d$ tends to zero. For the measures, the variance estimate gives $Q_{m,j}\to \mathbf1_{D_ c}$ away from $R=a$. The pushforward of $\nu _0$ by $R$ is $q(1-R)^{q-1}\, \mathrm{d}R$, by the radial computation in the proof of theorem 18.38.4 applied to $1-R$. In particular the level set $\{ R=a\} $ has zero volume. Since $q!A_ d/m^ q\to k^ q$ and these constants are bounded, dominated convergence proves total-variation convergence of $q!m^{-q}B_{m,j}^0\nu _0$ to $\mu _ c$, and