On the distributions of absorbed particles in crossing a field containing absorption points (Q2898450)

Let \(x\), \(q\neq 1\), \(t\) be real numbers, \(k\) an integer. Define \([x]_q= (1-q)^x)/(1-q)\), \([x]_{k,q}= [x]_q [x-1]_q\dots[x- k+1]_q\), \(k= 0,1,\dots\), NEWLINE\[NEWLINE{x\brack k}_q= {[x]_{k,q}\over [k]_q!},\quad k= 0,1,\dots,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\prod^n_{i=1} (1+ tq^{i-1})= \sum^n_{k=0} q^{{k\choose 2}}{n\brack k}_q t^k,\quad -\infty< t<\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\prod^n_{i=1} (1- tq^{i-1})^{-1}= \sum^\infty_{k=0} {n+k-1\brack k}_q t^k,\quad |t|< 1,\;0< q< 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEE_q(t)= \prod^\infty_{i=1} (1+ (1- q)q^{i-1} t)= \sum^\infty_{k=0} q^{{k\choose 2}}{t^k\over [k]_q!},\quad -\infty< t<\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINEe_q(t)= \sum^\infty_{i=1} (1- (1-q) q^{i-1} t)^{-1}= \sum^\infty_{k= 0} {t^k\over [k]_q!},\quad |t|< 1/(1- q),\quad 0< q< 1.NEWLINE\]NEWLINE Note that \(E_q(t)e_q(-t) =1\) and \(E_{1/q}(t)= e_q(t)\) for \(0< q< 1\) and \(\text{abs}(t)< 1/(1- q)\).NEWLINENEWLINEConsider a sequence of independent Bernoulli trials with odds of success at the \(i\)th trial given by NEWLINE\[NEWLINE\theta_i= \theta q^{i-1},\quad i= 1,2,\dots,\quad 0<\theta< \infty,\quad 0< q< 1.NEWLINE\]NEWLINE Let \(X_n=\) the number of successes in \(n\) trials, and \(T_k=\) the number of trials until the \(k\)th success. Then \(X_n\) follows the \(q\)-binomial distribution-1 given by: NEWLINE\[NEWLINE\operatorname{P}(X_n= k)= {n\brack k}_q {q^{{k\choose 2}}\theta^k\over \prod^n_{i=1} (1+ \theta q^{i-1})},\quad k= 0,1,\dots, n,NEWLINE\]NEWLINE and \(T_k\) follows the \(q\)-Pascal distribution-1 given by: NEWLINE\[NEWLINE\operatorname{P}(T_k= n)= {n-1\brack k-1}_q {\theta^k q^{{k\choose 2}+n-k}\over \prod^n_{i=1} (1+\theta q^{i-1})},\quad n= k,k+1,\dots.NEWLINE\]NEWLINE Further, if the conditional probability of failure at any trial given the occurrence of \(j\)-\(1\) failures in the previous trials is given by: NEWLINE\[NEWLINEp_j= 1-\theta q^{j-1},\quad j= 1,2,\dots,\quad 0<\theta< 1,\quad 0< q< 1.NEWLINE\]NEWLINE Then \(X_n\) follows the \(q\)-binomial distribution-2 given by: NEWLINE\[NEWLINE\operatorname{P}(X_n= k)= {n\brack k}_q \theta^k \prod^{n-k}_{i=1} (1-\theta q^{i-1}),\quad k= 0,1,\dots, n,NEWLINE\]NEWLINE and \(T_k\) follows the \(q\)-Pascal distribution-2 given by: NEWLINE\[NEWLINE\operatorname{P}(T_k= n)= {n-1\brack k-1}_q \theta^k q^{n-k} \prod^{n-k}_{i=1} (1-\theta q^{i-1}),\quad n= k,k+1,\dots.NEWLINE\]NEWLINE As \(n\to\infty\), the \(q\)-binomial distribution-1 tends to the Heine distribution given by: NEWLINE\[NEWLINE\operatorname{P}(X= k)= e_q(-\lambda) {q^{{k\choose 2}}\lambda^k\over [k]_q!},\quad k= 0,1,\dots,NEWLINE\]NEWLINE where \(0< q< 1\) and \(\lambda= \theta(1- q)\). Similarly, the \(q\)-binomial distribution-2 tends to the Euler distribution given by: NEWLINE\[NEWLINE\operatorname{P}(X= k)= E_q(-\lambda) {\lambda^k\over [k]_q!},\quad k= 0,1,\dots,NEWLINE\]NEWLINE where \(0<\lambda< 1/(1- q)\) (these details are found in [the author, J. Stat. Plann. Inference 140, No.~9, 2355--2383 (2010; Zbl 1201.60008)]).NEWLINENEWLINEThe author in the present paper proceeds to apply these results to the problem of ``distribution of absorbed particles'' first studied by \textit{S. Zacks} and \textit{D. Goldfarb} [``Survival probabilities in crossing a field containing absorption points'', Naval Res. Log. Quart. 13, No. 1, 35--48 (1966), \url{doi:10.1002/nav.3800130104}]: Consider a queue of particles that are required to cross a field containing a random number \(Y\) of traps acting independently. If a particle comes close to a trap it is ``absorbed'' with probability \(p>0\) with \(p+q=1\). Assume further that a trap can absorb no more than one particle. The sequential crossing of the field by the particles follows the model described above. Assuming that \(Y\) follows a Poisson distribution, Zacks and Goldfarb [loc.\,cit.]\ obtained the distribution of \(X_n\) in a form that is not very tractable. The present author obtains the following more tractable results.NEWLINENEWLINETheorem 3.1: Under the stated conditions,NEWLINE{\parindent=6.5mm \begin{itemize}\item[(a)] if \(Y\) follows a Heine distribution, then \(X_n\) follows a \(q\)-binomial-distribution-1 and \(T_k\) follows a \(q\)-Pascal-distribution-1,NEWLINE\item[(b)] if \(Y\) follows an Euler distribution, then \(X_n\) follows a \(q\)-binomial-distribution-2 and \(T_k\) follows a \(q\)-Pascal-distribution-2.NEWLINENEWLINE\end{itemize}}NEWLINETheorem 3.3: Under the stated conditions, if \(X_n\) follows a discrete distribution on the nonnegative integers and \(X_n\to X\) as \(n\to\infty\), then \(Y\) follows the same limiting distribution as \(X\).NEWLINENEWLINECorollary 3.4: Under the stated conditions:NEWLINENEWLINE(a) if \(X_n\) follows a \(q\)-binomial-distribution-1, then \(Y\) follows a Heine distribution,NEWLINENEWLINE(b) if \(X_n\) follows a \(q\)-binomial-distribution-2, then \(Y\) follows an Euler distribution.