A CLT for the L² modulus of continuity of Brownian local time

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Let {Lˣₜ; (x,t)∈ R¹× R¹₊} denote the local time of Brownian motion and αₜ:=∫₋∞∞ (Lˣₜ)² dx. Let η=N(0,1) be independent of αₜ. For each fixed t ∫₋∞∞ (Lˣ⁺ʰₜ- Lˣₜ)² dx- 4ht h³/² L→(64 3)¹/²√αₜ η, as h 0. Equivalently ∫₋∞∞ (Lˣ⁺¹ₜ- Lˣₜ)² dx- 4t t³/⁴ L→(64 3)¹/²√α₁ η, as t∞.