Asymptotics of solutions to the linear search problem

ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PR OBLEM R OBIN A. HEINONEN MaLGa & DICCA, University of Geno a, Geno a, IT Abstract. The exact leading asymptotics of solutions to the symmetric linear search problem are obtained for an y p ositive probabilit y density on the real line with a monotonic, sufficiently regular tail. A similar result holds for densities on a compact in terv al. 1. Introduction Ho w should one mov e on a line in order to find, in minim um exp ected time, an unseen target whose position was dra wn from a kno wn probabilit y densit y? This is the question asked b y the classical “linear search problem” (LSP) [10, 3, 13, 1], in tro duced independently b y R. Bellman and A. Bec k in the early 1960s. In the literature, a large share of the attention on the LSP has b een fo cused on the case where the probabilit y density is symmetric ab out the searcher’s starting p oin t. In such cases, it is not hard to see that the optimal tra jectory zigzags back and forth across the starting p oin t, so that the tra jectory may b e parametrized b y a discrete sequence of turning p oints { x k } with alternating sign. With this in mind and with no loss of generality , we will, throughout this article, denote the turning p oin ts by a sequence of nonne gative real num bers { x k } (i.e. the mo duli of the turning p oints), with the starting p oint x 0 set to 0. Searc hing under uncertaint y is a central task in many scien tific areas— biology , rob otics, computer science, and game theory , to name a few—and the LSP is one of the simplest, most fundamen tal searc h problems that can b e devised. 1 Ho w ever, despite its simplicit y , the LSP is more difficult than it may seem at first glance, and relativ ely little is kno wn in general about optimal searc h strategies. F or example, no kno wn algorithm can compute the optimal strategy for general probabilit y densities (although dynamic programming can b e used to compute an ϵ -optimal solution for an y desired accuracy ϵ [1]). Moreo v er, to our kno wledge, there are essen tially no quan titative results ab out the optimal strategies for general probabilit y densities. Most of what is known ma y b e summarized by a few sp ecial cases, studied by A. Beck (1984,1986) [5, 6]: • the uniform distribution on a symmetric, compact in terv al, wherein the optimal strategy is simply to visit the endp oin ts; Date : F ebruary 25, 2026. 1 The LSP is also v ery closely related to the 2-lane cow-path problem in computer science [16]. 1 2 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM • the triangular distribution on the same in terv al, whose optimal tra jectory nev er terminates (and for which some numerical v alues of the x k w ere com- puted); • and the normal distribution, for which we hav e x k ∼ √ 2 k log k . In the presen t w ork, we close this latter gap definitiv ely b y rigorously establishing asymptotics for the optimal turning points for any (even tually) monotonic, nonzero densit y on the real line, under reasonable regularity assumptions. The result ma y b e expressed compactly in terms of the hazard function asso ciated with the target probabilit y density . A similar result can b e used to ev aluate the asymptotics for man y densities on compact interv als. In the cases of b oth b ounded and unbounded domains, w e also establish rigorously the asymptotics of the turning p oints in the case where p decays lik e a p o w er law (up to undetermined constan ts), which will serv e as an imp ortant edge case. The remainder of the pap er is organized as follows. In Sec. 2, we state the LSP precisely along with some useful, w ell-kno wn results and a few necessary definitions. Then, in Sec. 3 w e state our main results. A few in teresting sp ecial cases are presen ted in Sec. 4, including the aforemen tioned triangular distribution and the normal distribution (as a sp ecial case of general stretched/compressed-exponential tails). Finally , the main results are pro ved in Secs. 5–9, and we conclude with a brief discussion in Sec. 10. 2. Preliminaries 2.1. Problem statement and kno wn results. F ormally , the symmetric LSP ma y b e stated as follows: Problem 1 (Symmetric linear searc h problem) . Fix a pr ob ability density p ( x ) on R such that p is even (i.e., p ( x ) = p ( − x ) for al l x ∈ R ). Sele ct a tar get x ∗ ∼ p. L et Γ 1 b e the set of c ontinuous, pie c ewise C 1 , unit-sp e e d curves γ : [0 , ∞ ) → R with γ (0) = 0 . Find γ ∗ := arg min γ ∈ Γ 1 E x ∗ ∼ p [inf { t : γ ( t ) = x ∗ } ] . Muc h of what we know ab out Problem 1 was established by A. Beck in a long series of colorfully-named pap ers [3, 4, 8, 9, 5, 6, 7]. 2 F or one, as previously stated, the candidate γ may b e parametrized b y a discr ete sequence of turning p oints { x k } k ≥ 0 , and WLOG w e can tak e x 0 = 0 and x k ≥ 0 for all k ≥ 1. W e also hav e the following imp ortant, well-kno wn facts, which we state without pro of: Prop osition 1. L et { x k } b e a se quenc e of turning p oints p ar ametrizing a minimiz- ing se ar ch str ate gy for Pr oblem 1. Then the fol lowing hold: (1) Such a minimizing str ate gy { x k } exists if and only if R ∞ 0 xp ( x ) dx < ∞ . (2) Define the survival function G ( x ) = R ∞ x p ( t ) dt. Then { x k } minimizes the obje ctive function J [ x ] := ∞ X k =1 x k ( G ( x k ) + G ( x k − 1 )) (Ob j) and in p articular we have (by differ entiating) the r e curr enc e ( x k + x k +1 ) p ( x k ) = G ( x k ) + G ( x k − 1 ) . (Rec) (3) We have x k +1 > x k ∀ k ≥ 0 and, if p > 0 everywher e, x k → ∞ as k → ∞ . 2 “The linear search problem: electric b oogalo o” w as a working title for the presen t manuscript. ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 3 The form of the ob jective function J follo ws from a straigh tforward computation of the exp ectation of the first passage time (note that a p -dep endent constant term has b een omitted). The recurrence (Rec) essentially reduces the computation of the turning p oints to finding x 1 , but this is known to b e extremely difficult. The third fact—that the optimal turning points are strictly increasing in mo dulus—is non trivial and very useful. It is also ob vious that an y optimal strategy must visit the en tirety of the support of p , or else the expected first passage time div erges; for example, for p > 0 on R , w e m ust hav e that x k → ∞ as k → ∞ . This latter case will be the main focus of what follows; due to symmetry , it will generally suffice to think of p as a densit y on R + . 2.2. The hazard function. W e find it useful to p erform our analysis in terms of the hazar d function , a standard ob ject in the theory of probability densities ov er R + and surviv al analysis [15]. Definition 1. L et p b e a pr ob ability density on R + , and let G ( x ) := R ∞ x p ( t ) dt b e its survivor function. The function h ( x ) := p ( x ) /G ( x ) is c al le d the hazard function of p. Equiv alently , if X is the random v ariable sp ecifying the (mo dulus of the) target p osition, we hav e h ( x ) = lim ϵ → 0 Pr( X ∈ [ x, x + ϵ ) | X ≥ x ) ϵ . That is, h ( x ) represents the instantaneous rate p er unit distance that the target is lo cated at p osition x, given that it is not lo cated closer to the origin than x. Notably , the hazard of a density p uniquely sp ecifies p ; in particular, since h = p/G = − G ′ /G, we hav e G ( x ) = exp( − R x 0 h ( t ) dt ) . W e present the hazards of a few w ell-known kinds of probability densities b elow. Example 1 (Pareto) . If p ( x ) ≍ x − α for some α > 1 , then the hazar d is h ( x ) ∼ ( α − 1) /x. Example 2 (Stretc hed/compressed exponential) . If p ( x ) ≍ exp( − ( x/a ) b ) for some a, b > 0 , then the hazar d is h ( x ) ≍ x b − 1 . In p articular, the exp onential distribution has c onstant hazar d, and the normal distribution has line ar hazar d. Example 3 (Lognormal) . If p ( x ) ≍ x − 1 exp( − log 2 x/ 2 σ 2 ) for some σ > 0 , then the hazar d is h ( x ) ≍ log x/x. Example 4 (Gumbel) . If p ( x ) ≍ exp( − exp( x )) , then h ( x ) ≍ exp( x ) . It is clear from the definition that the hazard must itself b e p ositiv e wherever p > 0 . How ever, for the kinds of distributions in whic h we are in terested (monotone, sufficien tly regular), m uc h more can be said. In particular, w e ha ve the follo wing useful Lemma, which establishes a sharp low er b ound on h. Lemma 1. L et p > 0 b e a pr ob ability density on R + such that R ∞ 0 xp ( x ) dx < ∞ . L et h b e the hazar d function asso ciate d with p. If h is eventual ly monotone and L := lim x →∞ xh ( x ) exists, then h ( x ) ≥ c/x for some c > 1 and lar ge enough x. Pr o of. If h is even tually increasing, say for all x > x 0 , then h ( x ) ≥ h ( x 0 ) > 0 and for any c > 0 we hav e x ≥ c/h ( x 0 ) for large enough x. Therefore h ( x ) ≥ c/x ev entu ally . 4 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM No w supp ose h is even tually decreasing. If L = ∞ , w e ha ve immediately that h ( x ) = ω (1 /x ) by definition. Otherwise take L ∈ (0 , ∞ ) . Then h ( x ) ∼ L/x and log G ( x ) = − Z x 0 h ( t ) dt = − L log x + o (log x ) so G ( x ) = x − L + o (1) . But the first moment is Z ∞ 0 xp ( x ) dx = Z ∞ 0 G ( x ) dx < ∞ so L > 1 . T aking any c ∈ (1 , L ) then gives the claim for large enough x. □ After dividing through by G ( x k ) and taking logarithms, the recurrence (Rec) can b e re-expressed in terms of the hazard as follows: log (( x k + x k +1 ) h ( x k ) − 1) = Z x k x k − 1 h ( x ) dx. (Rec’) Through this equation, controlling how, and by how muc h, the hazard can change o v er the in terv al [ x k − 1 , x k ] will pro vide p o werful control ov er the optimal turning p oin ts themselves. This motiv ates the use of monotonicity and regularity hypothe- ses on the hazard, whic h are weak assumptions in the sense that they apply to essen tially an y commonly studied distribution on R + with finite exp ectation. 2.3. Regular v ariation and de Haan’s class Γ . In order to control the in tegral on the RHS of (Rec’), we need the hazard to b e sufficiently regular. The precise necessary notions of regularity are standard [12] and stated b elow. 3 Definition 2 (Slow v ariation) . L et L : [ A, ∞ ) → R + for some A > 0 b e L eb esgue- me asur able. We say that L is slowly v arying if, for al l λ > 0 L ( λx ) ∼ L ( x ) as x → ∞ . Morally sp eaking, slo wly v arying functions are slow er than p olynomial; a pro- tot ypical example is any pow er of a logarithm. As a useful fact, they ob ey the so-called Potter b ound , i.e., for any η > 0 and t 1 , t 2 large enough, there exists C η suc h that L ( t 1 ) L ( t 2 ) ≤ C η max (  t 1 t 2  η ,  t 1 t 2  − η ) . Slo w v ariation forms the basis of the notion of “regular v ariation.” Definition 3 (Regular v ariation) . L et f : [ A, ∞ ) → R + for some A > 0 b e L eb esgue-me asur able. We say that f is regularly v arying of index ρ and f ∈ R ( ρ ) if f ( x ) = x ρ L ( x ) for some slow ly varying function L. 3 It is probably p ossible to generalize and streamline our results by using a unified notion suc h as “Beurling regular v ariation” [11], but instead we c ho ose to work with more w ell-kno wn notions. ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 5 In particular, an y function with a p ow er-law tail is regularly v arying. A standard, useful consequence of regular v ariation is the uniform c onver genc e pr op erty whic h sa ys that, for any λ > 0 , f ( λx ) f ( x ) → λ ρ uniformly in λ. Regular v ariation will help us c haracterize the solutions for polynomially gro wing or slow er hazard. In order to extend the results to rapidly v arying (sa y , exp onen- tially growing) functions, we also need an alternate kind of regularit y , which is also standard: Definition 4 (De Haan’s class Γ) . L et f : [ A, ∞ ) → R + for some A > 0 b e L eb esgue-me asur able. We say that f is in de Haan’s class Γ and f ∈ Γ if ther e exists an auxiliary function a ( x ) > 0 such that for al l fixe d t ∈ R , f ( x + ta ( x )) f ( x ) → e t as x → ∞ . As a remark, it is well-kno wn that an y suc h auxiliary function necessarily sat- isfies a ( x ) = o ( x ) (a fact which we will use more than once) and moreov er is self-ne gle cting , that is a ( x + ta ( x )) a ( x ) → 1 , uniformly in t. Note in particular that if f ( x ) = exp( g ( x )) , with g increasing and t wice-differen tiable and g ′′ /g ′ 2 → 0 , then f ∈ Γ with auxiliary 1 /g ′ . Hence, Γ encompasses essentially an y “nice” function growing faster than any p olynomial. Conv ersely , one can show that any f ∈ Γ grows faster than p olynomial (see pro of of Theorem 2). W e close this section with a useful fact: p inherits monotonicit y from h if h is either R V or in class Γ . Prop osition 2. L et p > 0 b e a density on R + with hazar d h. If h is eventual ly monotone and either r e gularly varying or in de Haan ’s class Γ , then p is eventual ly monotone de cr e asing. Pr o of. First consider h ∈ R ( ρ ) . First consider ρ < 0 , so that h is ev entually mono- tone decreasing. Note that p ( x ) = h ( x ) exp  − R x 0 h ( t ) dt  , so for any λ > 1 p ( λx ) p ( x ) = h ( λx ) h ( x ) exp − Z λx x h ( t ) dt ! ≤ h ( λx ) h ( x ) → λ ρ , with the last step coming from uniform conv ergence. Hence for any ϵ > 0 , p ( λx ) p ( x ) ≤ λ ϵ + ρ , and taking ϵ < | ρ | yields p ( λx ) p ( x ) < 1 for large enough x. No w consider ρ > 0 ( h even tually increasing). F rom the Potter b ound, we hav e for any λ > 1 and η > 0 , h ( λx ) h ( x ) ≤ C η λ ρ + η 6 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM for sufficiently large x. Moreov er, R λx x h ( t ) dt ≥ ( λ − 1) xh ( x ) . Com bining these, we ha v e p ( λx ) p ( x ) ≤ C η λ ρ + η e − ( λ − 1) xh ( x ) < 1 for large enough x. Finally , w e turn our attention to h ∈ Γ . In this case, h is even tually increasing, and we hav e for any t > 0 Z x + ta ( x ) x h ( u ) du ≥ ta ( x ) h ( x ) , so p ( x + ta ( x )) p ( x ) ≤ h ( x + ta ( x )) h ( x ) e − ta ( x ) h ( x ) → e t (1 − a ( x ) h ( x )) . It remains to show that a ( x ) h ( x ) → ∞ . F or any t > 0 , and ϵ ∈ (0 , 1) , h ( x + a ( x )) /h ( x ) ≥ e 1 − ϵ and (due to the self-neglecting property) a ( x + a ( x )) /a ( x ) ≥ 1 − ϵ for large enough x. Then, defining a sequence { x n } b y x n +1 = x n + a ( x n ) , w e ha ve a ( x n +1 ) h ( x n +1 ) a ( x n ) h ( x n ) ≥ (1 − ϵ ) e 1 − ϵ > 1 for small enough ϵ. Hence a ( x n ) h ( x n ) → ∞ . W e can extend this to an y x by c ho osing n suc h that x ∈ [ x n , x n +1 ] , which gives a ( x ) ≍ a ( x n ) (by self-neglecting) and hence a ( x ) h ( x ) ≳ a ( x n ) h ( x n ) → ∞ . □ 3. Resul ts W e first pro ve the following lemma, whic h essentially says that the solutions to the LSP cannot gro w faster than exponential under monotone hazards. T o establish this con trol, it is necessary to use the ob jective function itself (rather than just the recurrence). The pro of strategy , which inv olves introducing a “comp etitor” sequence y k , previously was used, e.g. in [5] when studying the sp ecial case of a normally distributed target. Lemma 2. L et { x k } b e a minimizing se quenc e of turning p oints for Pr oblem 1. Under the same hyp otheses on h and p as in L emma 1, ther e exists M > 1 such that x k +1 ≤ M x k for lar ge enough k . Pr o of. F or any sequence { y k } such that y k = x k ∀ k ≤ N , we may write J [ y ] − J [ x ] = X k ≥ N (( y k + y k +1 ) G ( y k ) − ( x k + x k +1 ) G ( x k )) ≤ ( y N + y N +1 ) G ( y N ) − ( x N + x N +1 ) G ( x N ) + X k ≥ N +1 ( y k + y k +1 ) G ( y k ) = ( y N +1 − x N +1 ) G ( x N ) + X k ≥ N +1 ( y k + y k +1 ) G ( y k ) . But by optimality of x k , J [ y ] ≥ J [ x ] , so x N +1 ≤ y N +1 + 1 G ( x N ) X k ≥ N +1 ( y k + y k +1 ) G ( y k ) . (1) ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 7 No w, by Lemma 1, there exists c > 1 and x 0 > 0 such that h ( x ) ≥ c/x for all x > x 0 . Then u ≥ v ≥ x 0 implies log G ( u ) G ( v ) = − Z u v h ( t ) dt ≤ − Z u v c t dt = − c log u v = ⇒ G ( u ) ≤ G ( v )  u v  − c . (2) T ake N large enough that x N ≥ x 0 , and set r = 2 1 /c ∈ (1 , 2) . Define y N + j = x N r j for j ≥ 1 . Then combining Eq. 1 and Eq. 2 yields x N +1 ≤ rx N + ∞ X j =1 x N r j (1 + r ) r − cj = r x N + x N (1 + r ) r c − 1 − 1 , so x N +1 ≤ M x N with M = ( r c + 1) / ( r c − 1 − 1) = 3 / (2 r − 1 − 1) ∈ (1 , ∞ ) . □ The Lemma can b e strengthened if w e kno w that h ( x ) = ω (1 /x ) . Corollary 1. If, in addition to the hyp otheses of L emma 2, we have h ( x ) = ω (1 /x ) , then for any M > 1 , ther e exists K such that x k +1 ≤ M x k whenever k ≥ K . Pr o of. Same as the pro of of Lemma 2, mutatis mutandis . □ W e no w turn our atten tion to the increments ∆ k := x k − x k − 1 . Under an ad- ditional weak regularity assumption on h, we can establish a trichotom y whic h classifies the limiting v alue of these increments (zero, finite, or infinite). Theorem 1. L et p > 0 b e a pr ob ability density on R + such that R ∞ 0 xp ( x ) dx < ∞ . L et h b e the hazar d function asso ciate d with p, and let { x k } b e a minimizing se quenc e of turning p oints for Pr oblem 1 (using the symmetric extension p ( x ) = p ( − x ) as the tar get density). Define ∆ k := x k − x k − 1 . If h is eventual ly monotone and L := lim x →∞ xh ( x ) exists, then the fol lowing hold: (1) If h ( x ) = o (log x ) , then ∆ k → ∞ . (2) If h ( x ) = ω (log x ) , then ∆ k → 0 . (3) If h ( x ) ∼ c log x for some c ∈ (0 , ∞ ) , then ∆ k → 1 /c. In particular, asymptotically logarithmic hazards—equiv alent to densities with tail p ≍ exp( − cx log x + O ( x ))—act as a b oundary case where the optimal turning p oin ts gro w linearly . This is useful for certain applications whic h in tro duce an inequalit y c onstrain t to the LSP , see for example [14]. While the incremen ts are in teresting ob jects in their o wn right, the cen tral result of this article is that, with a bit of extra regularit y on h (in the sense defined in the previous section), studying them in fact allows us to characterize the exact leading asymptotics of x k . W e m ust exclude the c ase of p ow er law tails ( h ( x ) ∼ 1 /x ), wherein LSP b ehav es qualitatively differently . This is summarized in the following Theorem. Theorem 2. Supp ose, in addition to the hyp otheses of The or em 1, that h ( x ) = ω (1 /x ) and either h ∈ R ( ρ ) for some ρ ∈ R or h ∈ Γ . Then ∆ k = log(2 x k h ( x k )) h ( x k ) + o  1 h ( x k )  (3) and k ∼ Z x k h ( x ) log( xh ( x )) dx. 4 (4) 4 W e are unable to characterize the error term in Eq. 4 without finer-grain con trol on h. 8 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM Theorem 2 characterizes the asymptotics of any (well-behav ed) density which deca ys faster than a p o w er law; for any explicit expression for h, w e can in prin- ciple compute the integral (4) and in v ert to find the leading b eha vior of x k . F or completeness, we also work out the case of p o wer-la w (Pareto) tails, in which case the optimal turning p oints grow exp onen tially . 5 Prop osition 3. Supp ose h ( x ) ∼ a/x with a > 1 , with h eventual ly monotone. Then under optimality, x k = r k + o ( k ) , wher e r is the unique solution to r a = a ( r + 1) − 1 (5) such that r > 1 . Mor e over, if h ( x ) = a/x + O ( x − (1+ ϵ ) ) for some ϵ > 0 , then x k ≍ r k . The pro of is deferred to Sec. 7. 3.1. Extension to compact interv als. If p is instead supp orted only on a com- pact interv al, say [ − 1 , 1] WLOG, the LSP may seem qualitatively differen t, at least prima facie . Ho wev er, a result closely analogous to Theorem 2 holds and charac- terizes the asymptotics as x → 1 for regular and monotonic hazards. A subtlety is that we must first establish that the sequence of optimal turning p oin ts do es not terminate; the question of whether or not the sequence terminates was already answ ered in Ref. [2] (indeed, in greater generality than sho wn here), but we presen t the following simple results for clarity . Prop osition 4. L et p > 0 b e a symmetric, c ontinuous pr ob ability density on ( − 1 , 1) such that lim x ↑ 1 p ( x ) = 0 . If p is monotone on some interval [ A, 1) with A ∈ (0 , 1) , then the se quenc e x k do es not terminate (i.e. x k < 1 ∀ k ). Pr o of. Suppose the contrary; then x N − 1 = x N = 1 for some N . Consider a second sequence { y k } terminating at N + 1 with y k = x k for 0 ≤ k ≤ N − 2 , y N − 1 = t > x N − 2 , and y N = y N +1 = 1 . Then J [ y ] − J [ x ] = ( t − 1) G ( x N − 2 ) + ( t + 1) G ( t ) ≥ 0 , so for any t ∈ [0 , 1) , G ( t ) / (1 − t ) ≥ G ( x N − 2 ) / (1 + t ) . But for t close enough to 1, p is monotonically decreasing and thus we can tak e G ( t ) / (1 − t ) ≤ p ( t ) and hence p ( t ) ≥ G ( x N − 2 ) / (1 + t ) > 1 2 G ( x N − 2 ) . But we can alwa ys take t close enough to 1 that this is false, contradiction. □ On the other hand, if p do es not go to 0 at the b oundary of the interv al, it is easy to show that the optimal search strategy reaches the b oundary in finite time, and it do es not make sense to talk ab out large- k asymptotics. Prop osition 5. L et p > 0 b e a symmetric, c ontinuous pr ob ability density on ( − 1 , 1) such that L := lim x ↑ 1 p ( x ) > 0 . Then ther e exists N ∈ N such that x N = 1 . Pr o of. Assume the contrary , so x k < 1 for all k . Then x k ↑ 1 as k → ∞ . Since p is con tin uous, G ( x ) = R 1 x p ( t ) dt → 0 . But ( x k + x k +1 ) p ( x k ) = G ( x k ) + G ( x k − 1 ) , and the LHS → 2 L > 0 while the RHS → 0 , contradiction. □ 5 This result is lik ely known, but it is unclear if it has ever been published. ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 9 If the hazard is sufficiently regular and go es to infinity at the b oundary suffi- cien tly quic kly , we hav e the following theorem, whic h w e pro ve in Sec. 8. Theorem 3. L et p > 0 b e a symmetric, c ontinuous pr ob ability density on ( − 1 , 1) such that lim x ↑ 1 p ( x ) = 0 . L et G ( x ) = R 1 x p ( t ) dt and h ( x ) = p ( x ) /G ( x ) . L et { x k } b e an optimal str ate gy for Pr oblem 1, and let ∆ k := x k − x k − 1 . If, on some interval [ A, ∞ ) with A > 0 , ˜ h ( x ) := h (1 − 1 /x ) is monotone and either r e gularly varying of index ρ > 1 or in de Haan ’s class Γ , then the asymptotics Eq. 3-4 hold. Once again, we find that h ( ϵ ) ≍ 1 /ϵ (with ϵ = 1 − x ) represents a b oundary case which must b e treated differently than the others. T o study this case, it is illuminating to first establish the following simple Lemma, the analog to Lemma 1: Lemma 3. L et p b e a c ontinuous and eventual ly monotone density on (0 , 1) with p → 0 as x ↑ 1 . Then the hazar d ob eys h ( x ) ≥ 1 / (1 − x ) for x ∈ (0 , 1) close enough to 1. Pr o of. Since p → 0 and is p ositiv e, p is monotone decreasing on [ A, 1) for some A > 0 . Then for all x ∈ [ A, 1) , G ( x ) = Z 1 x p ( t ) dt ≤ (1 − x ) p ( x ) and so h ( x ) = p ( x ) /G ( x ) ≥ 1 / (1 − x ) . □ Note the imp ortan t distinction: due to the fact that finite exp ectation is not a useful con trol for densities on compact interv als, we are unable to guarantee that h ( x ) ≥ c/ (1 − x ) with c > 1 strictly . In the ev ent that c > 1 , which corresp onds to densities going to 0 like a p o w er la w at the b oundary , we can prov e the follo wing prop osition, showing that the residual 1 − x k deca ys doubly exp onentially . Prop osition 6. L et p > 0 b e a symmetric, c ontinuous pr ob ability density on ( − 1 , 1) . If the hazar d h satisfies h ∼ c/ (1 − x ) as x → 1 for some c > 1 , then log(1 − x k ) ≍ −  c c − 1  k . If, furthermor e, h ( x ) = c/ (1 − x ) + O ((1 − x ) δ − 1 ) for some δ > 0 , then 1 − x k ∼ 2 c exp − A  c c − 1  k ! for some A > 0 . The pro of of Prop osition 6 is given in Sec. 9. How ever, the b oundary case where h ( x ) ∼ 1 / (1 − x ) remains difficult to describ e in generality; this corresp onds to p (1 − ϵ ) ∼ ℓ (1 /ϵ ) for some ℓ slowly v arying. In this case, the asymptotics of the optimal turning points app ear to depend sensitiv ely on the next-to-leading b ehavior of the hazard. W e defer the classification of suc h cases to future study . 4. Examples Eq. 4 allows us find the asymptotics of { x k } for essentially any well-behav ed, ev entu ally monotonic hazard on the real line, except for the degenerate case h ( x ) ≍ 1 /x (p o w er-la w tails), which w as handled in Prop osition 3. Here are a few inter- esting examples whic h do not app ear to b e known in the literature; they follow 10 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PROBLEM from straightforw ard calculations, whose details we hav e suppressed for the sake of brevit y . Example 5 (Compressed/stretc hed exp onen tial tails) . Supp ose, in addition to the hyp otheses of The or em 2, we have h ( x ) ∼ ax b for any a > 0 , b > − 1 . Then under optimality, x k ∼  1 + b a k log k  1 / (1+ b ) . Note that the ab ov e subsumes as s pecial cases b oth the exp onen tial ( b = 0) and normal ( b = 1) distributions; the asymptotics of { x k } in the latter case were one of the the main results of Ref. [5]. Example 6 (Lognormal tails) . Supp ose, in addition to the hyp otheses of The o- r em 2, we have h ( x ) ∼ a log x/x for any a > 0 . Then under optimality, x k ∼ e √ k log k/a . Example 7 (Gum b el tails) . Supp ose, in addition to the hyp otheses of The or em 2, we have h ( x ) ∼ e ax for a > 0 (Unlike the pr evious examples, this hazar d is de Haan Γ and not R V.) Then under optimality, x k ∼ 1 a log k . Let us also present tw o examples of a distribution on a compact interv al. First, consider the triangular distribution. It was previously shown in Ref. [5] that the optimal searc h strategy in this case has an infinite num ber of turning points, but no quantitativ e asymptotic was given. A quic k calculation giv es h ( x ) = 2 / (1 − x ) , and w e can apply Prop osition 6 to find that the distance to the interv al boundary deca ys lik e a double exp onential. Example 8 (T riangular distribution) . L et a symmetric pr ob ability density p > 0 b e supp orte d on ( − 1 , 1) and supp ose the hazar d ob eys h ( x ) ∼ 2 / (1 − x ) + O ((1 − x ) δ − 1 ) for some δ > 0 . Then under optimality, 1 − x k ∼ 4 e − A 2 k . for some A > 0 . Finally , we consider tails decaying lik e p (1 − ϵ ) ≍ exp( − a/ϵ c + o (1 /ϵ )) , c > 0 . Due to the rapid decay of p at the boundary , 1 − x k deca ys m uc h more slo wly in this case. Example 9 (A fast-decaying family of distributions on ( − 1 , 1)) . L et a symmetric pr ob ability density p > 0 b e supp orte d on ( − 1 , 1) and supp ose the hazar d ob eys h ( x ) ∼ a (1 − x ) 1+ b for some a, b > 0 . Then under optimality, 1 − x k ∼  1 + b a k  − 1 /b . 5. Proof of Theorem 1 Our proof strategy is similar to that of Lemma 2, in that we construct a com- p etitor sequence which upp er b ounds the growth of the incremen ts, this time in an h -dep enden t manner. ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 11 Pr o of. F rom (Rec), w e ha ve b y monotonicity that h k := min { h ( x k − 1 ) , h ( x k ) } ≤ R x k x k − 1 h ( x ) dx ∆ k ≤ max { h ( x k − 1 ) , h ( x k ) } := ¯ h k . (6) Th us w e hav e ∆ k ≥ log (( x k + x k +1 ) h ( x k ) − 1) ¯ h k ≥ log (2 x k h ( x k ) − 1) ¯ h k ≥ log ( x k h ( x k )) ¯ h k , (7) where the last inequality follows from Lemma 1. W e first focus on the case h ( x ) = o (log x ) . If x k ≥ 2 x k − 1 , then ∆ k = x k − x k − 1 ≥ x k → ∞ . On the other hand, if x k < 2 x k − 1 , then log x k = log x k − 1 + O (1) , so ¯ h k = o (log x k ) regardless of whether h is increasing or decreasing. Hence ∆ k ≥ log x k + O (1) o (log x k ) → ∞ . (8) This prov es the first case. No w supp ose that h is increasing, consisten t with h = Ω(log x ). Then using Eq. 1, u ≥ v implies G ( u ) = G ( v ) exp  − Z u v h ( x ) dx  ≤ G ( v ) e − h ( u )( u − v ) . (9) Set y N + j = x N + r j for each j ≥ 1 , with r = log( x N h ( x N )) /h ( x N ) . W e then hav e G ( y N + j ) G ( x N ) ≤ exp( − h ( x N ) r j ) = ( x N h ( x N )) − j . Inserting into Eq. 1, we find x N +1 − x N ≤ r + ∞ X j =1 (2 x N + 2 r j + r )( x N h ( x N )) − j = r + 2 x N + r x N h ( x N ) − 1 + 2 x N h ( x n ) r ( x N h ( x N ) − 1) 2 = r + O  1 h ( x N )  . But the choice of N w as arbitrary , so we ha v e (in com bination with Eq. 7 and using h monotonically increasing) log( x k h ( x k )) h ( x k ) ≤ ∆ k ≤ log( x k − 1 h ( x k − 1 )) + O (1) h ( x k − 1 ) . (10) The upp er and low er b ounds b oth tend to zero if h ( x ) = ω (log x ) . On the other hand, b y Lemma 2, log ( x k ) = log x k − 1 + O (1), so if h ( x ) ∼ c log x, the upp er and low er b ounds are b oth equal to 1 /c + O (log log x k / log x k ) , so it follows that ∆ k → 1 /c. □ 6. Proof of Theorem 2 Equations 6 and 10 are already strongly suggestive of the final result. The main ingredient that extra regularity provides is a guarantee that h ( x k − 1 ) ∼ h ( x k ) through either the uniform conv ergence prop erty of regular v ariation or the o ( x ) prop ert y of the auxiliary functions for de Haan’s class Γ . The v alidit y of integrating 12 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PROBLEM the increments (Eq. 4) follows by establishing that the increments do not gro w to o quic kly . Pr o of. Using Corollary 1 and h ( x ) = ω (1 /x ), we ha ve from (Rec’) that Z x k x k − 1 h ( x ) dx = log(2 x k h ( x k )) + o (1) . Since h is even tually monotone, we can use Eq. 6 to write log(2 x k h ( x k )) + o (1) ¯ h k ≤ ∆ k ≤ log(2 x k h ( x k )) + o (1) h k . (11) W e first show that ∆ k = o ( x k ) . If h is decreasing, then h k = h ( x k ) and Eq. 11 giv es ∆ k x k ≤ log( x k h ( x k )) + O (1) x k h ( x k ) → 0 since x k h ( x k ) → ∞ . On the other hand, with h increasing, Eq. 10 still holds and in particular ∆ k x k ≤ ∆ k x k − 1 ≤ log( x k − 1 h ( x k − 1 )) + O (1) x k − 1 h ( x k − 1 ) → 0 . Next, we show that if h ∈ R ( ρ ) is regularly v arying, then h ( x k ) ∼ h ( x k − 1 ) so that ¯ h k ∼ h k , yielding Eq. 3 from Eq. 11. By the uniform conv ergence prop erty of regularly v arying functions, we hav e for any u ( x ) = o ( x ) that h ( x + u ( x )) h ( x ) = h ( x (1 + u ( x ) /x )) h ( x ) →  1 + u ( x ) x  ρ → 1 . Using u ( x k ) = − ∆ k giv es h ( x k − ∆ k ) /h ( x k ) = h ( x k − 1 ) /h ( x k ) → 1 . No w w e pro v e Eq. 4 for regularly v arying hazard. Let f ( x ) = log ( xh ( x )) /h ( x ) . it is easy to see that f is regularly v arying of index − ρ. The uniform con vergence prop ert y then gives     f ( λx ) f ( x ) − λ − ρ     ≤ η uniformly for any η > 0 , for large enough x. It follows that there exists δ > 0 such that | 1 − λ | ≤ δ implies 1 − η ≤ f ( λx ) f ( x ) ≤ 1 + η . Putting x = x k and λ = t/x k for t ∈ [ x k − 1 , x k ] , we ha ve λ ∈ [1 − ∆ k /x k , 1] , so since ∆ k /x k → 0 we even tually hav e (1 − η ) f ( x k ) ≤ f ( t ) ≤ (1 + η ) f ( x k ) . It follows that ∆ k (1 + η ) f ( x k ) ≤ Z x k x k − 1 dt f ( t ) ≤ ∆ k (1 − η ) f ( x k ) . But ∆ k /f ( x k ) → 1 , so taking η → 0 , we find R x k x k − 1 dt f ( t ) → 1 . Let F ( x ) = R x dt f ( t ) . Then for any k 0 < k, F ( x k ) − F ( x k 0 ) = k − 1 X j = k 0 ( F ( x j +1 ) − F ( x j )) → k − k 0 , so F ( x k ) ∼ k . This is the desired result. ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 13 Finally , we mo ve to the case where h ∈ Γ . Let a b e the auxiliary function. First, w e claim that a ( x ) = ω  log h ( x ) h ( x )  . (12) Supp ose the con trary , and there is some C > 0 suc h that a ( x ) ≤ C log h ( x ) /h ( x ) for large enough x. By Def. 4 with t = 1 , we hav e h ( x + a ( x )) /h ( x ) → e. Define a sequence y n +1 = y n + a ( y n ) for n ≥ 0 . Iterating then gives h ( y n ) ∼ e n h ( y 0 ) . (13) Hence, a ( y n ) ≤ C log h ( y n ) h ( y n ) ∼ C n e n h ( y 0 ) , and it follows that P n ≥ 0 a ( y n ) < ∞ and therefore y n con v erges to a finite limit, con tradicting Eq. 13. On the other hand, we claim that h ( x ) /x m → ∞ for any m > 0 . Indeed, if h has auxiliary function a, for any t > 1 we ha ve h ( x + ta ( x )) ≥ e t h ( x ) for large enough x. Define y n +1 := y n + ta ( y n ) . Then h ( y n ) ≥ e t h ( y n − 1 ) ≥ · · · ≥ e nt h ( y 0 ) . But ta ( y n ) ≤ y n /m since a ( x ) = o ( x ) , so y n +1 ≤ (1 + 1 /m ) y n and hence y n ≤ x 0 (1 + 1 /m ) n . It follows that h ( y n ) y m n ≥ h ( y 0 ) y m 0  e t (1 + 1 /m ) m  n → ∞ since e t > e > (1 + 1 /m ) m for any m. The claim follows from h b eing monotonic. Hence, log x = o (log h ) and we can refine Eq. 12 further to a ( x ) = ω  log( xh ( x )) h ( x )  . (14) Next, note that for r > 0 Z x x − ra ( x ) h ( t ) dt = a ( x ) h ( x ) Z r 0 h ( x − sa ( x )) h ( x ) ds → a ( x ) h ( x ) Z r 0 e − s ds = (1 − e − r ) a ( x ) h ( x ) . W e now claim that ∆ k = o ( a ( x k )) . Otherwise, there is a subsequence x k j and r > 0 such that ∆ k j ≥ ra ( x k j ) . Then 1 a ( x k j ) h ( x k j ) Z x k j x k j − 1 h ( t ) dt ≥ Z x k j x k j − ra ( x k j ) h ( t ) dt ∼ 1 − e − r But combining Z x k x k − 1 h ( t ) dt ≤ log ( x k h ( x k )) + O (1) with Eq. 14 yields 1 a ( x k ) h ( x k ) Z x k x k − 1 h ( t ) dt → 0 , a contradiction. 14 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PROBLEM Th us, s etting t k = ∆ k /a ( x k ) = o (1) , we hav e h ( x k − 1 ) h ( x k ) = h ( x k − 1 + t k a ( x k )) h ( x k ) → 1 b y Def. 4. This means w e once again hav e Eq. 3. Finally , we hav e h ( x − u ( x )) h ( x ) → 1 uniformly whenever u ( x ) = o ( a ( x )). But a ( x ) = o ( x ) , so we further hav e log(( x − u ( x )) h ( x − u ( x ))) log( xh ( x )) → 1 and so f ( x − u ( x )) f ( x ) → 1 , with f ( x ) = log( xh ( x )) /h ( x ) as before. This allo ws us to reason similarly as before, so that Eq. 4 holds. □ 7. Proof of Proposition 3 W e first need to establish existence/uniqueness of the ro ot of Eq. 5. Then, regular v ariation is enough to establish exp onential growth up to a subleading error term; more precise control on the asymptotics of h in turn is enough to establish that this error term is in fact zero. Pr o of. First, we claim that Eq. 5 has a unique solution on (1 , ∞ ) . T o see this, write F ( x ) := ( x a + 1) /a − 1 − x, so that solutions satisfy F ( x ) = 0 . Then x > 1 implies F ′ ( x ) = x a − 1 − 1 > 0 , so F is strictly increasing on (1 , ∞ ) . Moreov e r, F (1) = 2(1 − a ) /a < 0 , while F ( x ) → ∞ for x → ∞ , so the claim follows. No w let r k = x k /x k − 1 . By Lemma 2, r k ∈ (1 , C ] for some C > 1 even tually . Because p is measurable and p > 0 , h is also measurable and h ∈ R ( − 1), so the uniform conv ergence prop erty of regularly v arying functions gives sup t ∈ [1 ,C ] t     h ( tx ) h ( x ) − 1 /t     = sup t ∈ [1 ,C ]     h ( tx ) a/ ( tx ) − 1     → 0 and thus sup x ∈ [ x k − 1 ,x k ] | h ( x ) − a/x | ≤ sup x ∈ [ x k /C,x k ] | h ( x ) − a/x | = o (1 /x k ) . Hence, Z x k x k − 1 h ( x ) dx = a Z x k x k − 1 dx x + o Z x k x k − 1 dx x k ! = a log r k + o (1) . (15) On the other hand, ( x k + x k +1 ) h ( x k ) − 1 = x k h ( x k )(1 + r k +1 ) − 1 = ( a + o (1))(1 + r k +1 ) − 1 = a (1 + r k +1 ) − 1 + o (1) , ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 15 where the last step used the b oundedness of r k +1 . Th us, from (Rec’) and Eq. 15, w e ha v e r k +1 = r a k + 1 a − 1 + o (1) = g ( r k ) + o (1) , (16) where we hav e defined g ( x ) = ( x a + 1) /a − 1 . Next, let m := lim inf k →∞ r k and M := lim sup k →∞ . By Lemma 2, M < ∞ . W e also hav e m > 1; if not, then there is a subsequence k j suc h that r k j → 1 , and Z x k j x k j − 1 h ( x ) dx ∼ a log r k j → 0 . (17) But then, using (Rec’) and xh ( x ) → a , exp Z x k x k − 1 h ( x ) dx ! = x k h ( x k )(1 + r k +1 ) − 1 → a (1 + r k +1 ) − 1 > log(2 a − 1) > 1 , con tradicting Eq. 17. No w, choose a subsequence k j with r k j → M . F rom Eq. 16, r k j +1 → g ( M ) . But lim sup j →∞ r j ≤ M , so g ( M ) ≤ M , and F ( M ) ≤ 0 . Since F is increasing and F ( r ) = 0 , we conclude M ≤ r. Similarly , along a subsequence with r k j → m, we get r k j +1 → g ( m ) and mutatis mutandis we conclude m ≥ r . Therefore, r k → r . Finally , since x k = x 0 Q k j =1 r j , 1 k log x k = 1 /k log x 0 + k X j =1 log r j → log r so x k = exp( k (log r + o (1))) = r k + o ( k ) . This prov es the first part. If h ( x ) − a/x = O ( x − (1+ ϵ ) ) , then Z x k x k − 1 h ( x ) dx = a log r k + O  x − ϵ k − 1  and ( x k + x k +1 ) h ( x k ) − 1 = a (1 + r k +1 ) − 1 + O ( x − ϵ k ) . Since r k +1 is even tually b ounded, we then ha ve log(( x k + x k +1 ) h ( x k ) − 1) = log( a (1 + r k +1 ) − 1) + O ( x − ϵ k ) and so a log r k = log( a (1 + r k +1 ) − 1) + O ( x − ϵ k − 1 ) , or r a k = a (1 + r k +1 ) − 1 + O ( x − ϵ k − 1 ) . Put G ( x ) := ( a (1 + x ) − 1) 1 /a and call the error term ξ k . W e ha ve G ′ ( x ) = ( a (1 + x ) − 1) 1 /a − 1 = H ( x ) a − 1 so G ′ ( r ) = r 1 − a ∈ (0 , 1) , so G is a contraction mapping. 16 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PROBLEM Since G ( r ) = r w e hav e r k − r = G ( r k +1 ) − G ( r ) + ξ k and so | r k − r | ≤ λ | r k +1 − r | + | ξ k | with λ ∈ (0 , 1) . It follows that | r k − r | ≤ λ n | r k + n − r | + n − 1 X j =0 λ j | ξ k + j | for each n ≥ 1 . T aking n → ∞ , we conclude | r k − r | ≤ ∞ X j =0 λ j | ξ k + j | . W e already show ed that lim inf r k > 1 , so put r k ≥ 1 + δ for some δ > 0 . Then x k + j ≥ x k (1 + δ ) j so | ξ k + j | ≤ C x − ϵ k (1 + δ ) − j ϵ for some finite C > 0 . It follo ws that P ∞ j =0 λ j | ξ k + j | ≤ C ′ x − ϵ k for some finite C ′ , and therefore | r k − r | = O ( x − ϵ k ) . Finally , we hav e x k r k = x 0 k Y j =1 r j r whic h conv erges iff P k j =1 log r j r con v erges. But log ( r j /r ) = ( r j − r ) /r + O (( r j − r ) 2 ) = O ( r j − r ) , and since | r j − r | = O ( x − ϵ k ) with x k + j ≥ (1 + δ ) j x k for large enough k , the series is summable. Hence x k ∼ C r k for some C . □ 8. Proof of Theorem 3 The pro of is essentially the same structurally as that of Theorem 2; once again, the main p oint is to establish that h ( x k ) ∼ h ( x k − 1 ) using the uniform conv ergence prop ert y , although the fine details differ. Pr o of. First, Prop osition 4 shows that the x k are non-terminating. No w let ϵ = 1 − x, z = 1 /ϵ. W e first pro ve that ∆ k / (1 − x k ) → 0 . Supp ose, on the contrary , that ∆ k ≥ θ ϵ k for some θ ∈ (0 , 1) along some subsequence. Then on that subsequence, x k − 1 = x k − ∆ k ≤ 1 − (1 + θ ) ϵ k , so Z x k x k − 1 h ( t ) dt ≥ Z 1 − (1+ θ ) ϵ k 1 − ϵ k h ( t ) dt = Z (1+ θ ) ϵ k ϵ k ˜ h (1 /s ) ds. Since ˜ h is monotone, we hav e for all s ∈ [ ϵ k , (1 + θ ) ϵ k ] that ˜ h (1 /s ) ≥ ˜ h  z k 1 + θ  , so Z x k x k − 1 h ( t ) dt ≥ θ ϵ k ˜ h  z k 1 + θ  , On the other hand, x k + x k +1 ≤ 2 , so (Rec’) gives Z x k x k − 1 h ( t ) dt ≤ log (2 h ( x k )) = log(2 ˜ h ( z k )) , ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 17 so log(2 ˜ h ( z k )) ≥ θ ˜ h ( z k / (1 + θ )) z k . (18) No w supp ose that ˜ h ∈ R ( ρ ) . Cho osing λ = 1 / (1 + θ ) > 0 , we ha v e ˜ h ( λz ) / ˜ h ( z ) → λ ρ as z → ∞ , and hence ˜ h ( z k / (1 + θ )) ∼ (1 + θ ) − ρ ˜ h ( z k ) . Using ˜ h ( z ) ∼ z ρ L ( z ) (with L slowly v arying), we then hav e z k log(2 ˜ h ( z k )) θ ˜ h ( z k / (1 + θ )) ∼ (1 + θ ) ρ z k log(2 ˜ h ( z k )) θ ˜ h ( z k ) = O log z k z ρ − 1 k L ( z k ) ! → 0 since ρ > 1 , contradicting Eq. 18. If instead ˜ h ∈ Γ , we again hav e ˜ h ( z ) /z m → ∞ for any m > 0 . This immediately implies that ˜ h ( z k / (1 + θ )) z k log ˜ h ( z k ) → ∞ , again contradicting Eq. 18. No w, by monotonicity , Eq. 6 again holds, and since log (( x k + x k +1 ) h ( x k ) − 1) = log(2 x k h ( x k )) + o (1) , combining with (Rec’) again shows that ∆ k = log(2 x k h ( x k )) + o (1) h ( x k ) as desired (in fact, the factor of x k in the logarithm can be neglected in this case since x k → 1). The pro of of Eq. 4 is essentially unchanged from the corresp onding part of the pro of of Theorem 2. □ 9. Proof of Proposition 6 The pro of is structurally similar to that of Prop osition 3: w e establish that G is regularly v arying, and then use this to deriv e a recurrence for the logarithm of 1 − x k , up to O (1) errors. More precise control on the higher-order asymptotics of h suffices to shrink this error to o (1) and refine our estimate of the x k . Pr o of. Let ϵ = 1 − x and ϵ k = 1 − x k . W e hav e d log G (1 − ϵ ) /dϵ = h (1 − ϵ ) . W rite h (1 − ϵ ) = c/ϵ + r ( ϵ ) with r ( ϵ ) = o (1 /ϵ ) as ϵ ↓ 0 . Then, integrating, we hav e log G (1 − ϵ ) = log K + c log ϵ + Z ϵ r ( t ) dt. for some K > 0 Hence G (1 − ϵ ) = ϵ c ℓ (1 /ϵ ) with ℓ ( t ) := K exp( R 1 /t r ( u ) du ) . W e claim that ℓ is slowly v arying. T o see this, note that for any λ > 0 , log ℓ ( λt ) ℓ ( t ) = Z 1 /λt 1 /t r ( u ) du. But r ( u ) = o (1 /u ) , so for any η > 0 , | r ( u ) | ≤ η /u for small enough u, so      Z 1 /λt 1 /t r ( u ) du      ≤ η | log λ | . 18 ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEAR CH PROBLEM T aking t → ∞ and η → 0 shows that log ℓ ( λt ) ℓ ( t ) → 0 . W e then hav e G ( x k − 1 ) G ( x k ) =  ϵ k − 1 ϵ k  c ℓ (1 /ϵ k − 1 ) ℓ (1 /ϵ k ) . and so, using x k , x k +1 → 1 and h ( x ) ∼ c/ (1 − x ) , (Rec’) gives  ϵ k − 1 ϵ k  c ℓ (1 /ϵ k − 1 ) ℓ (1 /ϵ k ) = 2 c ϵ k (1 + o (1)) . The Potter b ound then gives, for any η > 0 , ( c − η ) log ϵ k − 1 ϵ k + O (1) ≤ log 2 c ϵ k ≤ ( c + η ) log ϵ k − 1 ϵ k + O (1) , so η → 0 shows that log 2 c ϵ k = c log ϵ k − 1 ϵ k + O (1) , or, writing L k = − log ϵ k , L k = c c − 1 L k − 1 + O (1) . It follows that L k ≍  c c − 1  k for some A > 0 , proving the first claim. On the other hand, if r = O ( ϵ δ − 1 ) , then we can refine our estimate of G further to G (1 − ϵ ) = K ϵ c (1 + O ( ϵ δ )) . No w (Rec’) gives  ϵ k − 1 ϵ k  c = 2 c ϵ k (1 + o (1)) , whence L k = c c − 1 L k − 1 + log 2 c 1 − c + o (1) . It follows that there exists A > 0 such that L k = A  c c − 1  k − log 2 c + o (1) , pro ving the second claim. □ 10. Discussion T aken together, Theorems 1–3 and Prop ositions 3–6 classify and quantify the asymptotics of the solutions to the symmetric LSP for positive, monotonic target densities on b oth compact interv als and the real line. This solv es a ma jor comp o- nen t of the problem for most well-behav ed densities. Tw o imp ortant classes hav e not b een treated carefully: densities with strong, p ersisten t oscillations throughout the tail (or as x approaches the b oundary in the case of compact in terv als), and densities on compact in terv als whic h approach 0 like a slo wly v arying function. While w e do not attempt to study such cases presen tly , it seems plausible that a similar approac h to the one taken here could pro vide control ASYMPTOTICS OF SOLUTIONS TO THE LINEAR SEARCH PROBLEM 19 on the asymptotics in the case of oscillating densities, pro vided that h has known upp er and low er b ounds. An interesting direction for future research would b e to try to use the rigorous asymptotics as a large- k constraint in a computational algorithm for the optimal searc h strategy . Early efforts by the author in this direction hav e b een encourag- ing but hav e th us far failed to pro duce convincing evidence of robust con v ergence prop erties. A cknowledgments The author thanks An tonio Celani for some early discussions whic h help ed in- spire this work. 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