Non-Gaussian semi-stable laws arising in sampling of finite point processes (Q265290)

A finite point process is characterized by the distribution of the number of points (the size) of the process. In some applications, for example, in the context of packet flows in modern communication networks, it is of interest to infer this size distribution from the observed sizes of sampled point processes, that is, processes obtained by sampling independently the points of i.i.d. realizations of the original point process. A standard nonparametric estimator of the of the size distribution has already been suggested in the literature, and it has been shown to be asymptotically normal under suitable but restrictive assumptions. When these assumptions are not satisfied, it is shown in the paper that the estimator can be attracted to a semi-stable law. The assumptions are discussed in the case of several concrete examples. A major theoretical contributions of this work are new and quite general sufficient conditions for a sequence of i.i.d. random variables to be attracted to a semi-stable law.NEWLINENEWLINEThe next theorem (results concerning semi-stable domain of attraction) is the main result of this work.NEWLINENEWLINETheorem 3.1. Let \(W_q\) be an integer-valued random variable taking value in \(0,1,2,\cdots\) such that for all \(x>0\), NEWLINE\[CARRIAGE_RETURNNEWLINE \mathbf P\Biggl(\frac{W_q}{2} \geq x,~~W_q\text{ is even }\Biggr)= \sum_{n-[x]}^\infty\mathbf P\Biggl(\frac{W_q}{2}=n\Biggr) = h_1([x])e^{\nu [x]}, CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINE \mathbf P\Biggl(\frac{W_q-1}{2} \geq x,~~W_q~\text{ is odd }\Biggr)= \sum_{n-[x]}^\infty\mathbf P\Biggl(\frac{W_q-1}{2}=n\Biggr) = h_2([x])e^{\nu [x]}, CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\nu>0\); the functions \(h_1\) and \(h_2\) satisfy NEWLINE\[CARRIAGE_RETURNNEWLINE \frac{h_2(x)}{h_1(x)}\to c_1~\text{ as }x\to \infty, CARRIAGE_RETURNNEWLINE\]NEWLINE for some fixed \(c_1\geq 0\), and NEWLINE\[CARRIAGE_RETURNNEWLINE \frac{h_1(ax)}{h_1(x)}\to 1~\text{ as }x\to \infty,~a\to 1. CARRIAGE_RETURNNEWLINE\]NEWLINE Let also NEWLINE\[CARRIAGE_RETURNNEWLINE X=L\Biggl(e^{W_q}\Biggr)e^{\beta W_q}(-1)^{W_q}, CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\beta>0\) and \(L\) is a slowly varying function at \(\infty\) such that \(e^{n}\) is ultimately monotonically increasing. Suppose that NEWLINE\[CARRIAGE_RETURNNEWLINE \alpha:=\frac{n}{2\beta}<2. CARRIAGE_RETURNNEWLINE\]NEWLINE Then, \(X\) is attracted to the domain of a semi-stable distribution in the following sense. If \(X,X_1,X_2, \ldots\) are i.i.d. random variables, then as \(n\to \infty\), the partial sums NEWLINE\[CARRIAGE_RETURNNEWLINE \frac{1}{A_{k_n}}\Biggl(\sum_{j=1}^{k_n}X_j-B_{k_n}\Biggr) CARRIAGE_RETURNNEWLINE\]NEWLINE converge to a semi-stable distribution with NEWLINE\[CARRIAGE_RETURNNEWLINE k_n=\Biggl[\frac{e^{n-1}\nu}{h_1(n-1)}\Biggr],~~ A_{k_n}=L\Biggl(e^{2n-2}\Biggr)e^{2\beta(n-1)} CARRIAGE_RETURNNEWLINE\]NEWLINE and \(B_{k_n}\) given by NEWLINE\[CARRIAGE_RETURNNEWLINE B_{k_n}=k_n\int_{1/k_n}^{1-1/k_n}\inf_y\{F(y)>s\}ds. CARRIAGE_RETURNNEWLINE\]NEWLINE The limiting semi-stable distribution is non-Gaussian, and is characterized by NEWLINE\[CARRIAGE_RETURNNEWLINE \alpha=\frac{\nu}{2\beta}. CARRIAGE_RETURNNEWLINE\]