Abstract
In this chapter we present the so-called Quasi-Monte Carlo (QMC) method, which can be seen as a deterministic alternative to the standard Monte Carlo method: the pseudo-random numbers are replaced by deterministic computable sequences of \([0,1]^d\)-valued vectors which, once substituted mutatis mutandis in place of pseudo-random numbers in the Monte Carlo method, may significantly speed up its rate of convergence, making it almost independent of the structural dimension d of the simulation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
When \(d=1\), an easy way to construct this sequence is to consider the countable family of continuous piecewise affine functions with monotonicity breaks at rational points of the unit interval and taking rational values at these break points (and at 0 and 1). The density follows from that of the set \({\mathbb Q}\) of rational numbers. When \(d\ge 2\), one proceeds likewise by considering continuous functions which are affine on hyper-rectangles with rational vertices which tile the unit hypercube \([0,1]^d\). We refer to [45] for more details.
- 2.
The name of this theorem looks mysterious. Intuitively, it can be simply justified by the multiple properties established as equivalent to the weak convergence of a sequence of probability measures. However, it is sometimes credited to Jean-Pierre Portmanteau in the paper: Espoir pour l’ensemble vide, Annales de l’Université de Felletin (1915), 322–325. In fact, one can easily check that no mathematician called Jean-Pierre Portmanteau ever existed and that there is no university in the very small French town of Felletin. This reference is just a joke hidden in the bibliography of the second edition of [45]. The empty set is definitely hopeless...
- 3.
The distribution function \(F_{\mu }\) of a probability measure \(\mu \) on [0, 1] is defined by \(F_{\mu }(x)=\mu ([0,x])\). One shows that a sequence of probability measures \(\mu _n\) converges toward a probability measure \(\mu \) if and only if their distribution functions \(F_{\mu _n}\) and \(F_{\mu }\) satisfy that \(F_{\mu _n}(x)\) converges to \(F_{\mu }(x)\) at every continuity point \(x\!\in {\mathbb R}\) of \(F_{\mu }\) (see [45], Chap. 1).
- 4.
This means that if the rational scalars \(\lambda ^i\!\in {\mathbb Q}\), \(i=0,\ldots , d\) satisfy \(\lambda ^0+\lambda ^1\alpha ^1+\cdots +\lambda ^d\alpha ^d=0\) then \(\lambda ^0=\lambda ^1=\cdots =\lambda ^d=0\). Thus \(\alpha \!\in {\mathbb R}\) is irrational if and only if \((1,\alpha )\) are linearly independent on \({\mathbb Q}\).
- 5.
A signed measure \(\nu \) on a space \((X, \mathcal{X})\) is a mapping from \(\mathcal{X}\) to \({\mathbb R}\) which satisfies the two axioms of a measure, namely \(\nu (\varnothing )=0\) and if \(A_n\), \(n\ge 1\), are pairwise disjoint, then \(\nu (\cup _nA_n) =\sum _{n\ge 1} \nu (A_n)\) (the series is commutatively convergent hence absolutely convergent). Such a measure is finite and can be decomposed as \(\nu =\nu _1-\nu _2\), where \(\nu _1\), \(\nu _2\) are non-negative finite measures supported by disjoint sets, i.e. there exists \(A\!\in \mathcal{X}\) such that \(\nu _1(A^c)=\nu _2(A)=0\) (see [258]).
- 6.
In fact, its variation in the Hardy and Krause sense is not finite either.
- 7.
Every real number in [0, 1) admits a p-adic expansion \(x= \sum _{k\ge 1}\frac{x_k}{p^k}\), \(x_k \!\in \{0,\ldots , p-1\}\), \(k\ge 1\). If x is not a p-adic rational, this expansion is unique. If x is a p-adic rational number, i.e. of the form \(x=\frac{N}{p^r}\) for some \(r\!\in {\mathbb N}\) and \(N\!\in \{0, \ldots , p^r-1\}\), then x has two p-adic expansions, one of the form \(x= \sum _{k=1}^\ell \frac{x_k}{p^k}\) with \(x_\ell \ne 0\) and a second reading \(x= \sum _{k=1}^{\ell -1} \frac{x_k}{p^k} +\frac{x_\ell -1}{p^\ell } \sum _{k\ge \ell +1} \frac{p-1}{p^k}\). It is clear that if x is not a p-adic rational number, its p-adic “digits” \(x_k\) cannot all be equal to \(p-1\) for k large enough. By definition the regular p-adic expansion of \(x\!\in [0,1)\) is the unique expansion of x whose digits \(x_k\) will be infinitely often not equal to \(p-1\). The case of 1 is specific: its unique p-adic expansion \(1= \sum _{k\ge 1}\frac{p-1}{p^k}\) will be considered as regular. This regular expansion is denoted by \(x=\overline{0.x_1x_2\ldots x_k\ldots }^p\) for every \(x\!\in [0,1]\).
- 8.
Bertrand’s conjecture was stated in 1845 but it is no longer a conjecture since it was proved by P. Tchebychev in 1850.
- 9.
Let \((X,\mathcal{X}, \mu )\) be a probability space. A mapping \(T:(X,\mathcal{X})\rightarrow (X,\mathcal{X})\) is ergodic if
$$ \begin{array}{ll} (i)&{} \mu \circ T^{-1}= \mu \;\;{ i.e.}\;\;\mu \text { is invariant under}\; T,\\ \\ (ii)&{}\forall \, A\!\in \mathcal{X},\; T^{-1}(A)=A \Longrightarrow \mu (A)=0 \text { or }1. \end{array} $$Then Birkhoff’s pointwise ergodic Theorem (see [174]) implies that, for every \(f\!\in L^1(\mu )\),
$$ \mu (dx)\text {-}a.s.\qquad \frac{1}{n} \sum _{k=1}^n f(T^{k-1}(x))\longrightarrow \int _Xf\, d\mu . $$The mapping T is uniquely ergodic if \(\mu \) is the only measure satisfying T. If X is a topological space, \(\mathcal{X}=\mathcal{B}or(X)\) and T is continuous, then, for any continuous function \(f:X\rightarrow {\mathbb R}\),
$$ \sup _{x\in X} \left| \frac{1}{n} \sum _{k=1}^nf(T^{k-1}(x))- \int _Xfd\mu \right| \longrightarrow 0\quad \text {as}\quad n\rightarrow +\infty . $$In particular, it shows that any orbit \((T^{n-1}(x))_{n\ge 1}\) is \(\mu \)-distributed. When \(X=[0,1]^d\) and \(\mu =\lambda _d\), one retrieves the notion of uniformly distributed sequence. This provides a powerful tool for devising and studying uniformly distributed sequences. This is the case e.g. for Kakutani sequences or rotations of the torus.
- 10.
Namely that for every \(\varepsilon >0\), \(\lambda _d(\{u\!\in [0,1]^d\,:\, \mathrm{dist}(u,\partial C) <\varepsilon \})\le \kappa _{_C} \varepsilon \).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Pagès, G. (2018). The Quasi-Monte Carlo Method. In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-90276-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-90274-6
Online ISBN: 978-3-319-90276-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)