The main feature here is that one has ak+1/bk = > 1 for all 1 k < M. If we have Xi ρ ≤ Ii for i = 1,2,...M, then all of the observations X1, X2, ..., XM are selected, so we ∈ always have the inequality M Ik) 1(Nρ M). ∏1(Xk ∈ ≤ ≥ k=1 Finally, by the independence of the variables Xk, 1 k M and the strict ≤ ≤ monotonicity of F, we see that the expectation of the product is strictly positive. This gives us P(Nρ M) > 0 for all M 1. Since M is arbitrary, we see that Nρ is ≥ ≥ unbounded, and the proof of the theorem is complete. Complements to Classical Record Theory Here we consider the calculation of the expected number of selections where we assume that there was a selection made at zero” that had value x [0,1]. “time ∈ Formally, we modify the definition (2) by first setting = min{j : Xj τ1 ≥ ρx}. Next, for k 2 we define as before by setting = min{j : Xj and ≥ τk τk ≥ ρXτk 1}, − finally we set . In this notation, R´enyi’s classical formula for the expected number of records is , and the main goal of this section is to generalize this result in two ways. The immediate goal is to show that , and then in Theorem 12 we will get a closely related formula for E[Rnx(ρ)]. We begin by using first step analysis to get a useful recursion for the quantities gn,ρ(x) E[Rnx(ρ)] and gn(x) def= gn,1(x). def= Specifically, if we consider the first observation y = X1, then X1 is not accepted if y ≤ and this happens with probability On the other hand if y = X1 [ρx,1] we do ρx, ρx. ∈ accept X1, and accordingly we find the basic recurrence relation (44) For general (0,1), this equation offers considerable resistance; in essence, it is a ρ ∈ linearized non-autonomous pantograph equation in integrated form. Nevertheless,
The main feature here is that one has ak+1/bk = ρ > 1 for all 1 ≤ k < M. If we have Xi ∈ Ii for i = 1,2,...M, then all of the observations X1, X2, ..., XM are selected, so we always have the inequality M ∏1(Xk ∈ Ik) ≤ 1(Nρ ≥ M). k=1 Finally, by the independence of the variables Xk, 1 ≤ k ≤ M and the strict monotonicity of F, we see that the expectation of the product is strictly positive. This gives us P(Nρ ≥ M) > 0 for all M ≥ 1. Since M is arbitrary, we see that Nρ is unbounded, and the proof of the theorem is complete. Complements to Classical Record Theory Here we consider the calculation of the expected number of selections where we assume that there was a selection made at “time zero” that had value x ∈ [0,1]. Formally, we modify the definition (2) by first setting τ1 = min{j : Xj ≥ ρx}. Next, for k ≥ 2 we define τk as before by setting τk = min{j : Xj ≥ ρXτk 1}, and − finally we set . In this notation, R´enyi’s classical formula for the expected number of records is , and the main goal of this section is to generalize this result in two ways. The immediate goal is to show that , and then in Theorem 12 we will get a closely related formula for E[Rnx(ρ)]. We begin by using first step analysis to get a useful recursion for the quantities gn,ρ(x) def= E[Rnx(ρ)] and gn(x) def= gn,1(x). Specifically, if we consider the first observation y = X1, then X1 is not accepted if y ≤ ρx, and this happens with probability ρx. On the other hand if y = X1 ∈ [ρx,1] we do accept X1, and accordingly we find the basic recurrence relation (44) For general ρ ∈ (0,1), this equation offers considerable resistance; in essence, it is a linearized non-autonomous pantograph equation in integrated form. Nevertheless,
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