A non-integral-dimensional random walk (Q5893802)

For a given \(d\in (1,\infty)\) and a given distribution \(\{\eta_ j\}_ 1^{[d]+1}\), this paper defines a space-homogeneous and time- inhomogeneous random walk (R.W.) S, best thought of as linear interpolation of the walk familiar for integral dimensions. Its behaviour is as if it were a simple d-dimensional R.W: Analogues of local central limit theorems, zero-one laws, hitting probabilities, recurrence and transience, etc. are obtained. The definition of the R.W. \(S=\{S_ m\}\) is as follows: \[ S_ m=\sum^{m}_{1}X_ i;\quad P(X_ i=\pm e_ j)=2^{-1}\gamma_ j(i),\quad i=1,2,...,j=1,2,...,D+1,\quad D=[d], \] where \(e_ j=(0,...,0,1,0,...,0)\in Z^{D+1}\) ((0,...,0,1) j elements), \[ \gamma_ j(i)=\eta_ j+O(i^{\beta -1})\quad if\quad j=1,2,...,D,\quad and\quad =\eta_{D+1}\beta i^{\beta -1}\quad if\quad j=D+1, \] satisfying \(\sum^{D+1}_{1}\gamma_ j(i)=1\) \(\forall i\), and \(\beta =d-D\).