Local and uniform moduli of continuity of chi--square processes

2021/06/01 by Marcus, Michael B., Rosen, Jay
#60F15 #60G15 #60G17 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2106.00542

Abstract

Let η=\η(t);t∈ [0,1]\ be a mean zero continuous Gaussian process with covariance U=\U(s,t),s,t∈ [ 0,1]\, with U(0,0)>0. Let \η_i;i=1,…, k\ be independent copies of η and set Y_k(t)=∑_i=1^k η²_i(t), t∈ [ 0,1]. The stochastic process Y_k =\Y_k (t),t∈ [ 0,1] \ is referred to as a chi--square process of order k with kernel U. Let ϕ(t) be a positive function on [0,δ] for some δ>0. If \limsup_{t→ 0}( η(t)-η(0))/( ϕ(t) )=1 a.s., then for all integers k≥ 1, \limsup_{t→ 0} \fracY_k (t)-Y_k (0) ϕ(t) = 2 Y^1/2_k(0) a.s. Set σ²(u,v)=E(η(u)-η(v))²\quadand \widetildeσ²(x)=sup_{|u-v|≤ x}σ²(u,v). Assume that inf_{t∈ [0,1]}U(t,t)>0 and, lim_x→0\widetildeσ²(x)log 1/x =0. Let φ(t) be a positive function on [0,1]. Then if lim_{h→ 0}sup_\stackrel|u-v|≤ h u,v∈Δ( η(u)-η(v))/( φ(|u-v|) )=1 a.s. for all intervals Δ⊂ [0,1], it follows that for all intervals Δ⊂ [0,1] and all integers k≥ 1, lim_{h→ 0}sup_\stackrel|u-v|≤ h u,v∈Δ \fracY_k (u)-Y_k (v) φ(|u-v|) = 2 sup_u∈ΔY_k ^1/2(u), \hspace.2 ina.s.

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