Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

scholarly journals On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

2014 ◽  
Vol 51 (2) ◽  
pp. 483-491 ◽  
Author(s):  
M. V. Boutsikas ◽  
D. L. Antzoulakos ◽  
A. C. Rakitzis
Keyword(s):  
Joint Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Independent Interest ◽  
Exchangeable Random Variables ◽  
Stopped Sums ◽  
Independent And Identically Distributed

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

scholarly journals Asymptotic Law of thejth Records in the Bivariate Exponential Case

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Grine Azedine
Keyword(s):  
Exponential Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Asymptotic Independence ◽  
Record Times ◽  
Bivariate Exponential ◽  
Exponential Case ◽  
Independent And Identically Distributed

We consider a sequence(Xi,Yi)1⩽i⩽nof independent and identically distributed random variables with joint cumulative distribution H(x,y), which has exponential marginalsF(x)andG(y)with parameterλ=1. We also assume thatXi(ω)≠Yi(ω),∀i∈N, andω∈Ω. We denoteRk(j)k⩾1andSk(j)k⩾1by the sequences of thejth records in the sequences(Xi)1⩽i⩽n,(Yi)1⩽i⩽n, respectively. The main result of of the paper is to prove the asymptotic independence ofRk(j)k⩾1andSk(j)k⩾1using the property of stopping time of thejth record times and that of the exponential distribution.

scholarly journals On the Joint Distribution of Stopping Times and Stopped Sums in Multistate Exchangeable Trials

2014 ◽  
Vol 51 (02) ◽  
pp. 483-491
Author(s):  
M. V. Boutsikas ◽  
D. L. Antzoulakos ◽  
A. C. Rakitzis
Keyword(s):  
Joint Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Independent Interest ◽  
Exchangeable Random Variables ◽  
Stopped Sums ◽  
Independent And Identically Distributed

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let S T,i be the stopped sum denoting the number of appearances of outcome 'i' in X 1, …, X T , 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S T,0, S T,1, …, S T,m ). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (1) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (01) ◽  
pp. 98-107
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x 1, · ··, xθ– 1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ +1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's. In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

scholarly journals Posted Price Mechanisms and Optimal Threshold Strategies for Random Arrivals

2021 ◽  
Author(s):  
José Correa ◽  
Patricio Foncea ◽  
Ruben Hoeksma ◽  
Tim Oosterwijk ◽  
Tjark Vredeveld
Keyword(s):  
A Priori ◽  
Random Variables ◽  
Random Order ◽  
Random Variable ◽  
Finite Sequence ◽  
Optimal Threshold ◽  
Independent Random Variables ◽  
Price Mechanism ◽  
Posted Price ◽  
Independent And Identically Distributed

The classic prophet inequality states that, when faced with a finite sequence of nonnegative independent random variables, a gambler who knows the distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. In this work, we consider the situation in which the sequence comes in random order. We look at both a nonadaptive and an adaptive version of the problem. In the former case, the gambler sets a threshold for every random variable a priori, whereas, in the latter case, the thresholds are set when a random variable arrives. For the nonadaptive case, we obtain an algorithm achieving an expected reward within at least a 0.632 fraction of the expected maximum and prove that this constant is optimal. For the adaptive case with independent and identically distributed random variables, we obtain a tight 0.745-approximation, solving a problem posed by Hill and Kertz in 1982. We also apply these prophet inequalities to posted price mechanisms, and we prove the same tight bounds for both a nonadaptive and an adaptive posted price mechanism when buyers arrive in random order.

scholarly journals APPROXIMATE RESULTS FOR A GENERALIZED SECRETARY PROBLEM

2011 ◽  
Vol 25 (2) ◽  
pp. 157-169 ◽  
Author(s):  
Chris Dietz ◽  
Dinard van der Laan ◽  
Ad Ridder
Keyword(s):  
Optimal Policy ◽  
Random Order ◽  
Secretary Problem ◽  
Backward Induction ◽  
Accurate Approximation ◽  
Long Time

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b ≥ 1 is a preassigned number. It is known, already for a long time, that for the optimal policy, one needs to compute b position thresholds (for instance, via backward induction). In this article we study approximate policies that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n → ∞) results, which show that the double-level policy is an extremely accurate approximation.

scholarly journals The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽  
2021 ◽  
Author(s):  
Krzysztof Jasiński
Keyword(s):  
Random Variables ◽  
Continuous Case ◽  
Coherent System ◽  
Coherent Systems ◽  
Discrete Random Variables ◽  
Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

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