It is often of interest to condition on a singular event given by a random variable, e.g. $\{Y=y\}$ for a continuous random variable $Y$. Conditional measures with respect to this event are usually derived as a special case of the conditional expectation with respect to the random variables generating sigma algebra. The existence of the latter is usually proven via a non-constructive measure-theoretic argument which yields an only almost-everywhere defined quantity. In particular, the quantity $\mathbb E[f|Y]$ is initially only defined almost everywhere and conditioning on $Y=y$ corresponds to evaluating $\mathbb E[f|Y=y] = \mathbb E[f|Y]{Y=y}$, which is not meaningful because of $\mathbb E[f|Y]$ not being well-defined on such singular sets. This problem is not addressed by the introduction of regular conditional distributions, either. On the other hand it can be shown that the naively computed conditional density $f_{Z|Y=y}(z)$ (which is given by the ratio of joint and marginal densities) is a version of the conditional distribution, i.e. $\mathbb E[\{Z\in B\}|Y=y] = \int_B f_{Z|Y=y}(z) dz$ and this density can indeed be evaluated pointwise in $y$. This mismatch between mathematical theory (which generates an object which cannot produce what we need from it) and practical computation via the conditional density is an unfortunate fact. Furthermore, the classical approach does not allow a pointwise definition of conditional expectations of the form $\mathbb E[f|Y=y]$, only of conditional distributions $\mathbb E[\{Z\in B\}|Y=y]$. We propose a (as far as the author is aware) little known approach to obtaining a pointwise defined version of conditional expectation by use of the Lebesgue-Besicovich lemma without the need of additional topological arguments which are necessary in the usual derivation.
We characterize the properties of convexity, compactness and preservation of upper hemicontinuity for conditional expectations of correspondences. These results are then applied to obtain a necessary and sufficient condition for the existence of pure strategy equilibria in Bayesian games.
Generalized conditional expectations, optional projections and predictable projections of stochastic processes play important roles in the general theory of stochastic processes, semimartingale theory and stochastic calculus. They share some important properties with ordinary conditional expectations. While the characterization of ordinary conditional expectations has been studied by several authors, no similar work seems to have been done for these three concepts. This paper aims at undertaking this task by giving And$\hat{o}$-Douglas type characterization theorem for each of them.
We prove that the exponential distribution is the only one which satisfies a regression identity. This identity involves conditional expectation of the sample mean of record values given two record values outside of the sample.
A known property of conditional expectation is extended to the framework of Markov kernels. Its meaning in terms of densities is provided. Some examples located in the field of clinical diagnosis are presented to delimit the main result of the paper.
We derive upper bounds on the tail conditional expectation of binomial and Poisson random variables. Those upper bounds are subsequently employed to the problem of obtaining non-asymptotic lower bounds on the probability that the aforementioned random variables are significantly larger than their expectation.
In this paper, we define a dynamically consistent conditional G-expectation in space $\mathbb{L}^{p}$, and give the related stochastic calculus of Itô's type, especially get Itô's formula for a general $C^{1,2}$-function.
In this paper we extend the definition of time conditional G-expectations $\mathbb{\hat{E}}_{t}[\cdot]$ to a larger domain on which the dynamical consistency still holds. In fact we can consistently define, by taking the limit, the time conditional expectations for each random variable $X$ which is the downward limit (resp. upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(Ω)$. To accomplish this procedure, some careful analysis is needed. Moreover, we give a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting …
In this paper, we investigate the convergence of products of conditional expectation operators. We show that if $(Ω,\cal{F},P)$ is a probability space that is not purely atomic, then divergent sequences of products of conditional expectation operators involving 3 or 4 sub-$σ$-fields of $\cal{F}$ can be constructed for a large class of random variables in $L^2(Ω,\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(Ω,\cal{F},P)$ is a purely atomic probability space, then products of conditional expectation operators involving any finite set of su…
We present some regularity properties (convexity, weak/weak* compactness and preservation of weak/weak* upper hemicontinuity) for Bochner/Gelfand conditional expectation of Banach valued correspondences under the nowhere equivalence condition. These regularity properties for Bochner/Gelfand integral of Banach valued correspondences are obtained as corollaries. Similar properties for regular conditional distributions are also covered by the corresponding results for Gelfand conditional expectation of correspondences. We prove the necessity of the nowhere equivalence condition for any of these p…
We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$ the expected value is the orthogonal projection onto the constants. Previous work has shown that the local field version of $L^\infty$ is the appropriate counterpart of $L^2$, and so the expected value of a local field-valued random variable is defined to be its ``projection'' in $L^\infty$ onto the constants. Unlike the real case, t…
We give a variational formulation for $-\log\mathbb{E}_ν\left[e^{-f}|\mathcal{F}_t\right]$ for a large class of measures $ν$. We give a refined entropic characterization of the invertibility of some perturbations of the identity. We also discuss the attainability of the infimum in the variational formulation and obtain a Prékopa-Leindler theorem for conditional expectations.
This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.
We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L^2 space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectat…
We derive two conditional expectation bounds, which we use to simplify cryptographic security proofs. The first bound relates the expectation of a bounded random variable and the average of its conditional expectations with respect to a set of i.i.d. random objects. It shows, under certain conditions, that the conditional expectation average has a small tail probability when the expectation of the random variable is sufficiently large. It is used to simplify the proof that the existence of weakly one-way functions implies the existence of strongly one-way functions. The second bound relaxes th…
We consider the joint lower expectation of a finite-state imprecise stochastic process, defined using either the Ville-Vovk-Shafer natural extension or the Williams natural extension. In both cases, we show that it is continuous with respect to the pointwise convergence of non-decreasing sequences of real-valued functions $f_n$, $n\in\mathbb{N}_0$, where each $f_n$ is $n$-measurable. For the Ville-Vovk-Shafer natural extension, a similar result is shown to hold for non-increasing sequences, provided that they converge to a bounded function.
When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by actual observations of the response of the real system. In a probabilistic setting, Bayes's theory is the proper mathematical background for this identification process. The possibility of being able to compute a conditional expectation turns out to be crucial for this purpose. We show how this theoretical background can be used in an actual numerical procedure, …
In this paper, we introduce the concept of hyperbolic valued random variables, their expectation and moments. We develop the hyperbolic analogue of Binomial and Poisson distributions. We study some of the properties of expectation on the basis of decomposition of a hyperbolic number into idempotent components. Finally we define conditional expectation of hyperbolic valued random variable and study some of its basic properties. Our random variable can take values which are zero divisors and this is the important part of this study.
A necessary and sufficient condition on a sequence $\{\mathfrak{A}_n\}_{n\in \mathbb{N}}$ of $σ$-subalgebras that assures convergence almost every where of conditional expectations is given.
After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defi…
Let $(W,H,μ)$ be the classical Wiener space, assume that $U_\la=I_W+u_\la$ is an adapted perturbation of identity where the perturbation $u_\la$ is an $H$-valued map, defined up to $μ$-equivalence classes, such that its Lebesgue density $s\to \dot{u}_\la(s)$ is almost surely adapted to the canonical filtration of the Wiener space and depending measurably on a real parameter $\la$. Assuming some regularity for $u_\la$, its Sobolev derivative and integrability of the divergence of the resolvent operator of its Sobolev derivative, we prove the almost sure and $L^p$-regularity w.r. to $\la$ of the…
Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free semicircular family. We show that conditional expectations and conditional non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the random matrix models associated to $V$. Then by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^…
This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in R^d , whose components are nonnegative and have finite expectation. Pointwise convergence of max-CFs is shown to be equivalent with convergence with respect to the Wasserstein metric. The space of max-CFs is not closed in the sense of pointwise convergence. An inversion formula for max-CFs is established.
A non-Euclidean generalization of conditional expectation is introduced and characterized as the minimizer of expected intrinsic squared-distance from a manifold-valued target. The computational tractable formulation expresses the non-convex optimization problem as transformations of Euclidean conditional expectation. This gives computationally tractable filtering equations for the dynamics of the intrinsic conditional expectation of a manifold-valued signal and is used to obtain accurate numerical forecasts of efficient portfolios by incorporating their geometric structure into the estimates.
Let $(A,\mathscr{A},μ)$ and $(B,\mathscr{B},ν)$ be probability spaces, let $\mathscr{F}$ be a sub-$σ$-algebra of the product $σ$-algebra $\mathscr{A}\times\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\mathscr{F}$ defines a bounded linear operator from $L^p(μ;L^q(ν;X))$ onto $L^p_{\mathscr{F}}(μ;L^q(ν;X))$, the closed subspace in $L^p(μ;L^q(ν;X))$ of all functions having a strongly $\mathscr{F}$-measurable representative.
For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of t…
We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we give universal scaling asymptotics for the conditional zero distribution around p. The key tool is the c…
Consider a pair of random vectors $(\mathbf{X},\mathbf{Y}) $ and the conditional expectation operator $\mathbb{E}[\mathbf{X}|\mathbf{Y}=\mathbf{y}]$. This work studies analytic properties of the conditional expectation by characterizing various derivative identities. The paper consists of two parts. In the first part of the paper, a general derivative identity for the conditional expectation is derived. Specifically, for the Markov chain $\mathbf{U} \leftrightarrow \mathbf{X} \leftrightarrow \mathbf{Y}$, a compact expression for the Jacobian matrix of $\mathbb{E}[\mathbf{U}|\mathbf{Y}=\mathbf{…
In this paper, we derive a representation for the value process associated to the solutions of FBSDEs in a jump-diffusion setting under multiple probability measures. Motivated by concrete financial problems, the latter representations are then applied to devise a generalization of the change of numéraire technique allowing to obtain recursive pricing formulas in the presence of multiple interest rates and collateralization.
The pricing of Bermudan options amounts to solving a dynamic programming principle, in which the main difficulty, especially in high dimension, comes from the conditional expectation involved in the computation of the continuation value. These conditional expectations are classically computed by regression techniques on a finite dimensional vector space. In this work, we study neural networks approximations of conditional expectations. We prove the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximat…
This paper deals with multidimensional dynamic risk measures induced by conditional $g$-expectations. A notion of multidimensional $g$-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, homogeneity and translatability properties of multidimensional conditional $g$-ex…
We consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering a decomposition of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) conditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.
We consider a channel $Y=X+N$ where $X$ is a random variable satisfying $\mathbb{E}[|X|]<\infty$ and $N$ is an independent standard normal random variable. We show that the minimum mean-square error estimator of $X$ from $Y,$ which is given by the conditional expectation $\mathbb{E}[X \mid Y],$ is a polynomial in $Y$ if and only if it is linear or constant; these two cases correspond to $X$ being Gaussian or a constant, respectively. We also prove that the higher-order derivatives of $y \mapsto \mathbb{E}[X \mid Y=y]$ are expressible as multivariate polynomials in the functions $y \mapsto \mat…
A quantum probability measure is a function on a sigma-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but whose values are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper we take a step toward a better mathematical understanding of quantu…
Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in $L^1(Ω,{\mathcal A},μ)$, where $μ$ is a probability measure, new conditional versions…
In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles. After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN. In the process, we see PBN can be used …
We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex $N\times N$ Ginibre ensemble, conditional on an arbitrary number $k=1,2,\ldots$ of complex eigenvalues.These objects provide the simplest generalisation of the expectations of the diagonal overlap ($k=1$) and the off-diagonal overlap ($k=2$) considered originally by Chalker and Mehlig. They also appear naturally in the pro…
The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that if …
