Large deviations for the Bessel clock (Q5937015)

Let \(B_t\), \(t\geq 0\), be a \(d\)-dimensional Brownian motion with \(d> 2\) and \(B_0\neq 0\). Consider the Bessel process \(R_t=|B_t|\), where \(|\cdot|\) denotes the Euclidean norm, and the Bessel clock \(C_t= \int^t_0 ds/R^2_s\). It is proved that if \(t\to\infty\), then NEWLINE\[NEWLINES_t= C_t/\log t\to 1/(d- 2)\quad\text{a.s.}NEWLINE\]NEWLINE and in \(L_p\), NEWLINE\[NEWLINE{1\over 2}\sqrt {(d-2)^3\log t}\Biggl(S_t- {1\over d-2}\Biggr)@> D>> N(0,1)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE-{1\over\log t} \log P(S_t> x)\to [(d- 2)x- 1]^2/8x\quad\text{for }x> 1/(d- 2).NEWLINE\]NEWLINE Next, functional versions of the above LLN, CLT and LDP are obtained.