Let \(X\subseteq \mathbb{R}^d\) be a fixed subset, \(\mu\) a fixed probability measure on \(X\), \(B_\delta(x)\) the ball in \(X\) with radius \(\delta\) and centre \(x\), and \(q\geq 2\) an integer. The fractal dimension (Rényi moment dimension) \(D_q(\mu)\) of the measure \(\mu\) is defined by \[ D_q(\mu)= \lim_{\delta\to 0} \Biggl[\log \int_X(\mu(B_\delta(x)))^{q- 1} \mu(dx)\Biggr]\Biggl/(q- 1)\log\delta. \] When the observations \(x_1,x_2,\dots, x_n\) can be regarded as independent random variables with common distribution \(\mu\), it is not difficult to show that the naive estimate \[ \widehat D_q(\delta, n,X):= \log\Biggl[ \sum^n_{i=1} (N(B_\delta(x_i)))^{q- 1}/(N(X))^q\Biggr]\Biggl/(q- 1)\log\delta, \] where \(N(A)\) counts the number of observed points falling into the set \(A\), produces a consistent estimate of \(D_q(\mu)\) provided that \(\delta\) approaches zero sufficiently slow as the number \(n\) of observations increases. But it is not clear what is to be expected when the observations come from a point process with more complicated structure.
Consider a simple, marked point process on the product space \(\mathbb{R}\times \mathbb{R}^d\). The points of this process are pairs \((t_i,x_i)\), where we interpret \(t_i\) as time and \(x_i\) as location. Two possible situations are discussed. In the first of these, the observation region \(X\subset \mathbb{R}^d\) is regarded as being held fixed, while the observations accumulate during the time interval \((0,T)\). Suppose the process is stationary and ergodic in time, it is shown that, under certain regularity constraints, \(\min\{\widehat D_q(\delta(T), n(T), X),d\}\) is a mean-square consistent estimate of \(D_q(\Pi)\) as \(T\) tends to infinity. In this notation, \(n(T)\) is the total number of cumulated observations in the region \(X\) during the time interval \((0,T)\), \(\delta(T)\) is a function, which approaches zero sufficiently slow as \(T\) tends to infinity, and \(\Pi\) is the stationary spatial distribution within \(X\) of the given point process. The same is true for a modification \(\widehat D_{[q]}\) of the estimate \(\widehat D_q\) using the factorial product counting measure \(N^{[k]}(\cdot)\) instead of the counting measure \(N(\cdot)\).
In the second case, the time \(T\) is held fixed, while the process is regarded as being stationary and ergodic in its spatial component and the observed region \(X\) is embedded in a sequence of incresing regions \(X_n\subset \mathbb{R}^d\). In that case it is shown that, under certain conditions, \(\min\{\widehat D_{[q]}(\delta(n), T,X_n),d\}\) is a mean-square consistent estimate of \(D^*_{[q]}\) as \(n\) tends to infinity, where \(D^*_{[q]}\) denotes the factorial Palm growth exponent of order \(q\) of the projection of the given point process on \((0,T)\times \mathbb{R}^d\) on \(\mathbb{R}^d\).