On Strassen-type results for the increments of two-parameter Wiener processes (Q2725214)

Assume that \(\{W(x,y):(x,y) \in[0,\infty) \times[0, \infty)\}\) is a two-parameter Wiener process defined on a probability space \(\{\Omega,{\mathcal F},P\}\). For a rectangle \(R(s,t,u,v)= [s,s+t]\times [u,u+v]\subset [0,\infty)\times [0,\infty)\), define NEWLINE\[NEWLINEW\bigl(R(s,t,u,v) \bigr)=W(s+t, u+v)-W(s,u+v)-W(s+t,u)+ W(s,u)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE{\mathfrak R}_T(s,x,t,y)= {W\bigl(R (s,a_Tx,t,b_Ty) \bigr)\over \sqrt {a_Tb_T}\beta_T},NEWLINE\]NEWLINE where \(0\leq s\leq A_T-a_T\) and \(0\leq t\leq B_T-b_T\) for some positive \(A_T\), \(B_T\), \(a_T\) \((\leq A_T)\) and \(b_T\) \((\leq B_T)\), which are functions of \(T\), \((x,y)\in[0,1]^2\) and NEWLINE\[NEWLINE\beta_T= \biggl\{2\bigl( \log(G_T)+ \log\log \bigl(\max(A_T,e) \bigr)+ \log\log \bigl(\max (B_T,e)\bigr) \biggr \}^{1/2}NEWLINE\]NEWLINE with \(G_T=[\max ((A_T-a_T)/a_T,1)]\) \([\max((B_T- b_T)/b_T,1)]\). The set of limit points of NEWLINE\[NEWLINE\bigcup_{T\geq 3}\bigl\{{\mathfrak R}_T (s,x,t,y): 0\leq s\leq A_T-a_T,\;0\leq t\leq B_T-b_T,\;(x,y)\in [0,1]^2\bigr\}NEWLINE\]NEWLINE is investigated. A major consequence of the main theorem is the following result for the increments of the two-parameter Wiener processes under some conditions for \(A_T\), \(B_T\), \(a_T\) and \(b_T\): NEWLINE\[NEWLINE\limsup_{T \to\infty} \sup_{0\leq s\leq A_T-a_T}\sup_{0\leq t\leq B_T-b_T} {\biggl|W\bigl(R (s,a_T,t,b_T) \bigr)\biggr |\over \sqrt{a_Tb_T} \beta_T}=1, \text{ a.s.}.NEWLINE\]