Manuscript Psy02-A@1 · Psy02-A.tex · theorem · d92ce402

Theorem: Time-uniform observable concentration

Environmental dynamics and the two-budget principle › A conditional time-uniform concentration budget · lines 957–988 · proof included or linked

Psy02-A@1#thm:time-concentration, Psy02-A@1#eq:time-budget

Under Assumption ass:environment,

n_{ϵ} (s) \leq A (s) n_{ϵ} (0) + B (s) .

K_{T} = s \leq T, j sup \norm v_{j} (s) < \infty, n_{ϵ, T} = s \leq T sup {A (s) n_{ϵ} (0) + B (s)},

then

s \leq T, j sup Tr (ρ_{ϵ, s} (O_{j, ϵ} (s) - o_{j} (s) 1)^{2}) \leq ϵ K_{T}^{2} (4 n_{ϵ, T} + 2) .

The bound vanishes if $ϵ n_{ϵ, T} \to 0$ .

Proof. Multiply the moment inequality by $A (s)^{- 1}$ and integrate. Apply Theorem thm:concentration at each time.

Displayed formulas in this passage

Psy02-A@1#eq:time-budget · line 971
$s \leq T, j sup Tr (ρ_{ϵ, s} (O_{j, ϵ} (s) - o_{j} (s) 1)^{2}) \leq ϵ K_{T}^{2} (4 n_{ϵ, T} + 2) .$
unlabeled · line 960
$n_{ϵ} (s) \leq A (s) n_{ϵ} (0) + B (s) .$
unlabeled · line 964
$K_{T} = s \leq T, j sup \norm v_{j} (s) < \infty, n_{ϵ, T} = s \leq T sup {A (s) n_{ϵ} (0) + B (s)},$