69 research outputs found
Prophet inequalities for i.i.d. random variables with random arrival times
Suppose are i.i.d. nonnegative random variables with finite
expectation, and for each , is observed at the -th arrival time
of a Poisson process with unit rate which is independent of the sequence
. For , 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 such that is
measurable with respect to the -algebra generated by
. For instance, it is shown that , where satisfies ; and this bound is asymptotically sharp as .
Another result is that , and this bound is
asymptotically sharp as . Upper bounds for the difference
are also given, under the additional assumption that the 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 is the distance from to
the nearest integer, and is an integer with . It is shown, by
using properties of a symmetric correlated random walk, that almost all level
sets of are finite (with respect to Lebesgue measure on the range of
), but that for an abscissa chosen at random from , the level set
at level is uncountable almost surely. As a result, the occupation
measure of is singular.Comment: 17 pages. An extra figure was added and several of the proofs are now
worked out in more detai
Differentiability and H\"older spectra of a class of self-affine functions
This paper studies a large class of continuous functions
whose range is the attractor of an iterated function
system 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 is
completely classified in terms of the contraction ratios of the maps
. Generalizing results of Lax (1973) and Okamoto (2006), it is
shown that either (i) is nowhere differentiable; (ii) is
non-differentiable almost everywhere but with uncountably many exceptions; or
(iii) 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 is determined.Comment: 41 pages; slightly restructured the proof of Theorem 6.1 and fixed a
few typo
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
, 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 ,
where , includes the continuous nowhere differentiable functions of
Perkins () and Bourbaki/Katsuura (), as well as the Cantor
function (). The main purpose of this article is to characterize the set
of points at which has an infinite derivative. We compute the Hausdorff
dimension of this set for the case , and estimate it for .
For all , we determine the Hausdorff dimension of the sets of points where:
(i) ; and (ii) has neither a finite nor an infinite derivative.
The upper and lower densities of the digit in the ternary expansion of
play an important role in the analysis, as does the theory of
-expansions of real numbers.Comment: 26 pages; more figures were added and Theorem 2.6 now includes
additional statement
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
A general "bang-bang" principle for predicting the maximum of a random walk
Let be either a Bernoulli random walk or a Brownian
motion with drift, and let , .
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
adapted to the natural filtration of , and is a nonincreasing convex
function. The optimal stopping time is shown to be of "bang-bang"
type: if the drift of the underlying process is
negative, and 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
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