证明
Lemma 3 ([A weighted quantile inequality]). For all sufficiently large odd m ,…Lemma 2 ([Hopf-weight distribution]). For m=2s-1 and 0≤b≤l , the eigenvalue…观点
the link. Choose an odd m≥7 satisfying Lemma 3 ([A weighted quantile inequality]), and fix 2 m-1 <c< J_m. For a small positive parameter ε , set q_*=1+ε, α_*^2=1-cε, a_*^2=q_*^ 1/m α_*^2. For q≥1 , the right-hand side of is F(q)=q^ 1/m (1-(2(q-1))/(m-1)), F'(q)=(m+1)/(m(m-1))q^ 1/m-1 (1-2q)<0. At q=q_* , the required strict inequality follows from c>2/(m-1) ; it consequently holds throughout [1,q_*] . In particular a_*^2<F(q_*)<1 . means. By, the eigenvalues at the dwell parameters are the numbers α_*^ -2 [B_l- (1-1 q_* )(2b-l)^2] with the multiplicities in Lemma 2 ([Hopf-weight distribution]). Thus their increasing order corresponds to the decreasing order of the Hopf weights (2b-l)^2 . If d(d+m-1)=λ and d≥0 , then 0≤√λ-d≤(m-1)/2 . Since B_l/l^2→1 , Lemma 2 ([Hopf-weight distribution]) and convergence of quantiles to a continuous, strictly increasing distribution imply d_ l, vN_l l ⟶ f_ε(v):=(1-cε)^ -1/2 √ 1-(1-1 q_* )y_m(v) (0<v<1). These rescaled exponents are uniformly bounded, so averaging the first uN_l terms gives, for every 0<u<1 , 1 l uN_l ∑_ i=1 ^ uN_l d_ l,i ⟶ g_ε(u):=1u∫_0^u f_ε(v)\,dv. One may see this directly by integrating the empirical quantile in ; the possible last fractional term has size O(N_l^ -1 ) after division by l . Define I(ε):=∫_0^1 g_ε(u)^ -m \,du. All functions here are bounded away from zero for sufficiently small ε . Uniformly for 0<v<1 , f_ε(v) =1+ε2(c-y_m(v))+O(ε^2). The same uniform remainder remains valid after averaging over (0,u) . Expansion of g_ε(u)^ -m and integration therefore give I(ε) , =1+(mε)/(2) [∫_0^11u∫_0^u y_m(v)\,dv\,du-c] +O(ε^2); , =1+(mε)/(2)( J_m-c) +O(ε^2)>1 all sufficiently small ε>0. The second equality uses the nonnegative integral identity ∫_v^1du/u=(1/v) . Fix such an ε , as small as desired, from now on. at integer thresholds. For each positive integer k and l≥2 , let M_l(k) be the largest initial group of the ordered exponents whose mean is strictly less than k , taking M_l(k)=0 if no group qualifies. Only finitely many degrees contribute: the displayed eigenvalue formula bounds all λ_ l,i below by a positive constant times l^2 , so _i d_ l,i grows at least linearly in l . The initial-group means are nondecreasing, and hence M_l(k)=N_l∫_0^1 1 1 uN_l ∑_ i=1 ^ uN_l d_ l,i <k \,du. For fixed u∈(0,1) and δ>0 , bounds the mean in this indicator by l(g_ε(u)+δ/2) for every sufficiently large l . It is therefore strictly below k whenever l≤k/(g_ε(u)+δ) , including at the upper endpoint. Using , the sum of the corresponding N_l , divided by ∑_ l=0 ^kN_l , has limit (g_ε(u)+δ)^ -m as the integer k→∞ . Letting δ0 and applying Fatou's lemma to yields _ k→∞\ k∈ℕ ∑_ l=2 ^∞M_l(k) ∑_ l=0 ^kN_l ≥∫_0^1g_ε(u)^ -m \,du =I(ε)>1. Discarding the finitely many lower degrees in this argument changes no limit. Consequently every sufficiently large integer k has a strict surplus. Taking L to be the largest degree with M_l(k)>0 and setting M_l=M_l(k) proves.