Prophet inequalities for i.i.d. random variables with random arrival times

Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time $S_k$ of a Poisson process with unit rate which is independent of the sequence $\{X_k\}$. For $t>0$, comparisons are made between the expected maximum M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)] and the optimal stopping value V(t):=\sup_{\tau\in\TT}\sE[X_\tau \sI(S_\tau\leq t)], where \TT is the set of all \NN-valued random variables $\tau$ such that $\{\tau=i\}$ is measurable with respect to the $\sigma$-algebra generated by $(X_1,S_1),...,(X_i,S_i)$. For instance, it is shown that $M(t)/V(t)\leq 1+\alpha_0$, where $\alpha_0\doteq 0.34149$ satisfies $\int_0^1(y-y\ln y+\alpha_0)^{-1} dy=1$; and this bound is asymptotically sharp as $t\to\infty$. Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is asymptotically sharp as $t\downarrow 0$. Upper bounds for the difference $M(t)-V(t)$ are also given, under the additional assumption that the $X_k$ are bounded.Comment: 16 pages with 1 figure; submitted to Sequential Analysis in shortened for

On the level sets of the Takagi-van der Waerden functions

This paper examines the level sets of the continuous but nowhere differentiable functions \begin{equation*} f_r(x)=\sum_{n=0}^\infty r^{-n}\phi(r^n x), \end{equation*} where $\phi(x)$ is the distance from $x$ to the nearest integer, and $r$ is an integer with $r\geq 2$. It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of $f_r$ are finite (with respect to Lebesgue measure on the range of $f$), but that for an abscissa $x$ chosen at random from $[0,1)$, the level set at level $y=f_r(x)$ is uncountable almost surely. As a result, the occupation measure of $f_r$ is singular.Comment: 17 pages. An extra figure was added and several of the proofs are now worked out in more detai

Predicting the supremum: optimality of "stop at once or not at all"

Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when the reward for stopping with a stopping time tau<=T is a nonincreasing convex function of M_T-X_tau. Under fairly general conditions on the process X_t, it is shown that the optimal stopping time tau is of "bang-bang" form: it is either optimal to stop at time 0 or at time T. For the case of random walk, the rule tau=T is optimal if the steps of the walk stochastically dominate their opposites, and the rule tau=0 is optimal if the reverse relationship holds. For Le'vy processes X_t with finite Le'vy measure, an analogous result is proved assuming that the jumps of X_t satisfy the above condition, and the drift of X_t has the same sign as the mean jump. Finally, conditions are given under which the result can be extended to the case of nonfinite Le'vy measure.Comment: 20 pages; added a few specific examples and additional reference

Level sets of signed Takagi functions

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as signed Takagi functions. The case when r_n=1 for all n is the classical Takagi function, a well-known example of a continuous but nowhere differentiable function. For f of the above form, the maximum and minimum values of f are expressed in terms of the sequence {r_n}. It is then shown that almost all level sets of f are finite (with respect to Lebesgue measure on the range of f), but the set of ordinates y with an uncountably large level set is residual in the range of f. The concept of a local level set of the Takagi function, due to Lagarias and Maddock, is extended to arbitrary signed Takagi functions. It is shown that the average number of local level sets contained in a level set of f is the reciprocal of the height of the graph of f, and consequently, this average lies between 3/2 and 2.Comment: This is a stand-alone version of Section 5 in arXiv:1102.1616, with more proof details. 15 pages, 2 figures. An error in the proof of Theorem 1.3 was corrected, and the theorem now has a slightly stronger statemen

Correction and strengthening of "How large are the level sets of the Takagi function?"

The purpose of this note is to correct an error in an earlier paper by the author about the level sets of the Takagi function [Monatsh. Math. 167 (2012), 311-331 and arXiv:1102.1616], and to prove a stronger form of one of the main results of that paper about the propensity of level sets containing uncountably many local level sets.Comment: 6 pages, correction of arXiv:1102.1616. A small mistake in the proof of Theorem 8 was correcte

The infinite derivatives of Okamoto's self-affine functions: an application of beta-expansions

Okamoto's one-parameter family of self-affine functions $F_a: [0,1]\to[0,1]$, where $0<a<1$, includes the continuous nowhere differentiable functions of Perkins ($a=5/6$) and Bourbaki/Katsuura ($a=2/3$), as well as the Cantor function ($a=1/2$). The main purpose of this article is to characterize the set of points at which $F_a$ has an infinite derivative. We compute the Hausdorff dimension of this set for the case $a\leq 1/2$, and estimate it for $a>1/2$. For all $a$, we determine the Hausdorff dimension of the sets of points where: (i) $F_a'=0$; and (ii) $F_a$ has neither a finite nor an infinite derivative. The upper and lower densities of the digit $1$ in the ternary expansion of $x\in[0,1]$ play an important role in the analysis, as does the theory of $\beta$-expansions of real numbers.Comment: 26 pages; more figures were added and Theorem 2.6 now includes additional statement

A general "bang-bang" principle for predicting the maximum of a random walk

Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible.Comment: 13 page

Digital sum inequalities and approximate convexity of Takagi-type functions

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1. Several inequalities are derived for S_b(N). Some of the inequalities can be interpreted as comparing the average value of s_b(n) over integer intervals of certain lengths to the average value of a beginning subinterval. Two of the main results are applied to derive a pair of "approximate convexity" inequalities for a sequence of Takagi-like functions. One of these inequalities was discovered recently via a different method by V. Lev; the other is new.Comment: 15 page

Hausdorff dimension of level sets of generalized Takagi functions

This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Takagi function is a special case. Sharp upper bounds are given for the Hausdorff dimension of the level sets of functions in these two families. Furthermore, the case where a function f is chosen at random from either family is considered, and results are given for the Hausdorff dimension of the zero set and the set of maximum points of f.Comment: 34 pages, 5 figures. The statement of Theorem 1.1 was expanded and various improvements to the presentation were mad

Differentiability and H\"older spectra of a class of self-affine functions

This paper studies a large class of continuous functions $f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes such classical examples as P\'olya's space-filling curves, the Riesz-Nagy singular functions and Okamoto's functions. The differentiability of $f$ is completely classified in terms of the contraction ratios of the maps $S_1,\dots,S_{m}$. Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) $f$ is nowhere differentiable; (ii) $f$ is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) $f$ is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of $f$ is determined.Comment: 41 pages; slightly restructured the proof of Theorem 6.1 and fixed a few typo