Manuscript En02-E@2 · En02-E-v2.tex · proposition · c2769986

Proposition: Tail estimates

Deterministic flagged Fock approximation › Tails and discarded moments · lines 616–649 · proof included or linked

En02-E@2#prop:tails

For every $N \geq 0$ ,

t_{N} (K) \leq min {1, μ / (N + 1)} .

For $λ_{m a x} > 0$ ,

t_{N} (K) \leq 1 < s < 1 + λ_{m a x}^{- 1} in f s^{- (N + 1)} G_{K} (s) .

The discarded mean occupation

m_{N} (K) = E [N_{tot} 1_{{N_{tot} > N}}] = μ - n = 0 \sum^{N} n π_{n} (K)

satisfies

m_{N} (K) \leq \frac{μ ^{2} + μ + tr K ^{2}}{N + 1}, m_{N} (K) \leq s^{- (N + 1)} s G_{K}^{'} (s)

on the same positive $s$ -domain. All these tails vanish for $K = 0$ .

Proof. Use Markov's inequality, then $1_{{n > N}} \leq s^{n - (N + 1)}$ . For the first moment, use $n \leq n^{2} / (N + 1)$ on the tail and $E N_{tot}^{2} = μ^{2} + μ + tr K^{2}$ . The derivative bound follows from the differentiated convergent generating series.

Displayed formulas in this passage

unlabeled · line 619
$t_{N} (K) \leq min {1, μ / (N + 1)} .$
unlabeled · line 623
$t_{N} (K) \leq 1 < s < 1 + λ_{m a x}^{- 1} in f s^{- (N + 1)} G_{K} (s) .$
unlabeled · line 628
$m_{N} (K) = E [N_{tot} 1_{{N_{tot} > N}}] = μ - n = 0 \sum^{N} n π_{n} (K)$
unlabeled · line 633
$m_{N} (K) \leq \frac{μ ^{2} + μ + tr K ^{2}}{N + 1}, m_{N} (K) \leq s^{- (N + 1)} s G_{K}^{'} (s)$

Nearby in this section

section: Tails and discarded moments