Some limit theorems in geometric processes (Q1432853)

Let \(Y_1,Y_2,\dots\) be i.i.d. nonnegative random varibles and \(X_i= a^{1-i}Y_i\), \(S_0= 0\), \(S(n)= X_1+\cdots+ X_n\), \(n\geq 1\), \(N(t)= \sup\{n: S(n)\leq t\}\), \(A(t)= t- S(N(t))\), \(B(t)= S(N(t)+ 1)- t\). The paper derives identities, inequalities and asymptotic behaviour for \(Ea^{-N(t)}\) and \(ES(N(t)+ 1)\). For \(0< a< 1\), when \(Y_1\) is ``new better than used in expectation'' there are asymptotic inequalities for the expectations of \(S(N(t)+ 1)\), \(A(t)\), \(B(t)\), \(a^{-N(t)}\), \(\sum_{n\leq N(t)}a^{-n}\) that become equalities for exponential \(Y_1\). Identities connecting the distributions of the above random variables to the corresponding ones for \(a=1\) are derived. Inequalities between corresponding random variables for \(a=1\) and \(a\neq 1\) in terms of ``stochastically larger'' are proved. Applications in reliability are referred to.