Exponential tail bounds for max-recursive sequences

Exponential tail bounds are derived for solutions of max-recursive equations and for max-recursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of divide and conquer algorithms, in parallel search algorithms or in the height of random tree models. For the proof we determine asymptotic bounds for the moments or for the Laplace transforms and apply a characterization of exponential tail bounds due to Kasahara (1978).


Introduction
Stochastic recursive equations of max-type arise in a great variety of problems with a recursive stochastic component as in the probabilistic analysis of algorithms or in combinatorial optimization problems. For a list of examples in this area see the survey paper of Aldous and Bandyopadhyay (2005). We consider in this paper random sequences (X n ), satisfying recurrences of the type which fit with the general divide and conquer paradigma. The I (n) r ∈ {0,. . . ,n − 1} are subgroup sizes of the K subproblems in which a problem of size n is split, b r (n) are random toll terms arising from the splitting process, A r (n) is a random weighting term for subproblem r and (X (r) n ) are independent copies of (X n ) describing the parameter of the r-th subproblem. It is assumed that (X (1) n ), . . . , (X (K) n ), (A(n) = (A 1 (n), . . . , A K (n)), I (n) = (I (n) 1 , . . . , I (n) K ), b (n) = (b 1 (n), . . . , b r (n))) are independent while the coefficients and subgroup sizes A(n), I (n) , b (n) may be dependent. d = denotes equality in distribution. X n coincides with the maximum (worst case) of the weighted parameters of the subproblems 1, . . . , K in distribution. A general distributional limit theorem for this type of max-recurrences was given in Neininger and Rüschendorf (2005) , see also [9], [10] by means of the contraction method. The limit of X n after normalization is characterized as unique solution of a fixpoint equation of the form where (A r , b) are limits in L 2 of the coefficients (A r (n), b r (n)) and X r are independent copies of X.
In the present paper we study some conditions on the coefficients such that the normalized recursive sequences for some scaling sequences s n have exponential tails. We also give conditions which imply exponential tails of solutions of the fixpoint equation (2). In section 2 we consider the case of solutions X of the max-recursive equation in (2). For this case some existence and uniqueness results have been obtained in [8], [10]. Some results on tail bounds in particular for the worst case of FIND equation have been given in Grübel and Rösler (1996), Devroye (2001) and Janson (2004). We derive bounds for the moments of X and obtain by Kasahara's theorem (1978) exponential tail bounds for X. In section 3 we consider the case of max recursive sequences (X n ). We establish various conditions which imply bounds for the asymptotics of moments and Laplace transforms which again lead by Kasahara's theorem to exponential tail bounds for max-recurrences (X n ). As example we discuss the worst case of FIND sequence.

Exponential tail bounds for max-recursive equations
Exponential tail bounds are not easily directly accessible for max-recursive sequences (X n ) as in (1) or for solutions of max-recursive equations in (2). It will however turn out to be possible to get suitable bounds on the moments or on the Laplace transforms of (X n ) resp. X. These imply exponential tail bounds by the following lemma, which is a consequence of a more general theorem of Kasahara (1978).
We define for functions f, g on and Lemma 2.1 (Kasahara (1978)) Let X be a random variable p, a > 0 2) For general X and p ≥ 1 a) is further equivalent to: c) ln Ee tX ≤ as ct q (8) where c = q −1 (pa) −(q−1) and 1 p + 1 q = 1, 3) The statements in 1),2) remain valid also if ≤ as is replaced by asymptotic equivalence ∼ as .
Remark: In the paper of Kasahara (1978) the statement of Lemma 2.1 was given for the asymptotic equivalence case (as in part 3)). The method of proof in that paper however also allows to cover the ≤ as -bounds as in parts 1), 2) of Lemma 2.1.

Theorem 2.2 Consider the max recursive equation (2) and assume that E
then (2) has a unique solution X in M q 0 , the class of all distributions with finite moments of order q 0 . Further X has moments of any order and Proof: The existence and uniqueness of a solution X (in distribution) of the max recursive equation (2) follows from Neininger and Rüschendorf (2005), Theorem 5. For the proof of (9) we establish bounds for the moments of X: This implies that Thus the tail estimate follows from an application of Lemma 2.1.
Remark: Condition (9) can arise in various different ways. Two typical of these are Condition (12) implies in particular that sup then (10)  Condition b) allows more general toll terms b r but puts a stronger condition on the weights A r .
Example 2.1 Let X be the unique solution of the following max-recursive equation with finite moments of any order where [2], [4]). This fixpoint equation characterizes the limit of the worst case of FIND. Then and thus by Theorem 2.2, (see (10)) x for x ∈ (0, 1) and 0 < 1 q ln q < 1, this implies for q ≥ 3 (1)).
Therefore, Lemma 2.1 implies the tail estimate with a = e−1 e 2 . Remark: Exponential tail bounds for the limit X of the normalized worst case of FIND algorithm X n = Tn n were established in Grübel and Rösler (1996) and in Devroye (2001) and in a similar way as above in Janson (2004). Theorem 2.2 generalizes these bounds to a general class of max recursive equations. The normalized worst case of FIND algorithm X n is stochastically majorized by X i.e.
(see Devroye (2001)). Therefore, the exponential tail bounds hold also for the normalized worst case of FIND algorithm X n = Tn n uniformly in n ∈ N.

Tail bounds for max-recursive sequences
In this section we establish tail bounds for max-recursive sequences (X n ) as in (1). A central limit theorem for these kind of recursions has been given in a recent paper of Neininger and Rüschendorf (2005). We assume in the following version that equation (1) is already in stabilized form.
Theorem 3.1 (Limit theorem for max-recursive sequences, see [8]) Let (X n ) be a maxrecursive sequence as in (1) with X n ∈ L s , ∀n and assume the following conditions: 1. stabilization: Then (X n ) converges in distribution to a limit X * . Further, l s (X n , X * ) → 0 and X * is the unique solution of the recursive equation (2) in M s , the class of distributions with finite s-th moments.
In the following theorem we supplement this limit theorem by giving tail bounds for (X n ). For the proof we establish uniformly in n ∈ N bounds on the asymptotics of the moments and then obtain exponential tail bounds by Lemma 2.1. Let s n = s(n) be a monotonically nondecreasing norming sequence of X n and consider the normalized sequence Theorem 3.2 Let (X n ) be a max-recursive sequence as in (2), let p > 0 and let Y i r ≤ as cr 1/p as r → ∞ for i = 0, . . . , n 0 −1. Further we assume Then there exists some function h(x), such that and where a = 1 2e(max{ηb, e}) p in case a) and a = 1 2e(max{b, e}) p in case b).
Thus we obtain exponential tail bounds of order p uniformly for all Y n . Proof: We establish by induction uniformly in n ∈ N 0 moment estimates for (Y n ) which by Lemma 2.1 correspond to the exponential tail bounds in (23). For n = 0, . . . , n 0 −1 these bounds are given by assumption. The normalized sequence (Y n ) satisfies the modified recursive equation We denote by Υ n the distribution of (A(n), I Then conditioning by the vector (A(n), I (n) , b (n) ) we obtain in the induction step with β : In case b) we obtain For the proof of ( * ) we consider The terms cl 1 p + o(1) as q → ∞ are independent of n ∈ N o and thus we obtain from Lemma 2.1 a tail bound uniformly in n ∈ N as in (23).
Remark: To estimate the crucial term in (18) one can use the bound Further, | b r (n)| ≤ r |b r (n)| , which however is only a good estimate if one of the |b r (n)| is big while all other |b r (n)| are small. Without toll terms the following improved conditions yield subgaussian tail bounds.
Theorem 3.3 Let (X n ) be a max-recursive sequence as in (1) with zero toll terms, b r (n) = 0, r = 1, . . . , K and X n ≥ 0, ∀n. We assume Proof: The proof is by induction. For i ≤ n 0 − 1 (31) holds by assumption (29). For the induction step we obtain by conditioning as in section 2 for x ≥ 1 With Υ n = P (A(n),I (n) ) we thus obtain from majorized convergence By induction in q ≥ 1 and partial integration we obtain To deal with the (1 + o(1)) term let we obtain by some calculation and thus we obtain (1 + o(1)).
Thus at the induction step it is possible to choose the same constants c, C.
The bounds in the following theorem are based on the Laplace transform and allow unbounded toll terms.
Theorem 3.4 Let (X n ) be a max-recursive sequence as in (1) and let for some nondecreasing sequence s n ≥ 1 and q > 1 EX n = µs q n + r n , r n = o(s q n ).
For the induction step let Υ n denote the distribution of (I (n) , b (n) , A(n)). By conditioning and using the induction hypothesis we obtain as in the proof of Theorem 3.