Theses

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Large-scale Asymptotics in Fixed-sample and Sequential Multiple Testing
(Indian Statistical Institute, 2026-08-26) Roy, Rahul
The advent of large-scale data acquisition technologies has led to the routine emergence of massive, dynamically evolving datasets. This has created a pressing need for statistical methodologies capable of operating effectively in both static and sequential data environments. Multiple hypotheses testing, a central tool in such settings, must therefore be adapted to both fixed-sample and sequential frameworks to ensure effective decision-making while controlling error rates. This thesis comprises two main parts. The first part addresses fixed-sample multiple testing under dependence in a Bayesian framework. Building upon the two-group mixture normal model of Bogdan et al. (2011), we extend their methodology to exchangeable multivariate normal test statistics, thereby accommodating realistic dependency structures frequently encountered in high-dimensional applications. We derive sufficient conditions under which multiple testing procedures satisfy the Asymptotic Bayes Optimality under Sparsity (ABOS) property in the presence of dependence. Furthermore, we show that several classical procedures, including those of Bonferroni (1936), Šidák (1967), and Benjamini and Hochberg (1995), retain the ABOS property under suitable sparsity and dependence assumptions. The second part focuses on large-scale multiple testing in sequential settings, where data vectors or multiple data streams are observed over time rather than being available in full initially. Motivated by diverse applications, we investigate two distinct sampling termination schemes: (i) synchronous termination, in which sampling and testing for all hypotheses stop simultaneously; and (ii) asynchronous termination, in which different hypotheses may stop at different times. For the synchronous case, we develop the Oracle Intersection (OI) test, based on the local false discovery rate statistic under a two-group mixture model. The OI test guarantees exact control of both the False Discovery Rate (FDR) and the False Non-Discovery Rate (FNR) at prespecified levels. We further propose a fully datadriven version that achieves asymptotic simultaneous control of FDR and FNR as the number of hypotheses m → ∞. While existing sequential tests use fixed stopping boundaries, the proposed tests employ stopping boundaries that adapt quickly with the samplesize, ensuring shrinkage of the continue-sampling region as the trial progresses. As a result, stopping times for both the oracle and data-driven tests converge to a finite constant as m → ∞. Moreover, the ratio of the expected sample size of the OI test to that of the Gap rule (He and Bartroff, 2021) converges to zero as m →∞. Extensive analyses of real and simulated datasets demonstrate the superiority of the proposed methods over existing approaches. Finally, we address the case of asynchronous termination and introduce the Oracle Stagewise (OS) test, constructed from the local false discovery rate statistic under a two-group mixture model. Employing adaptive stopping boundaries, the OS test drops hypotheses by accepting or rejecting them at interim stages of sampling until all hypotheses are decided. This stagewise procedure achieves simultaneous control of the FDR and FNR, while substantially reducing the total sample size relative to existing sequential methods (Bartroff and Song, 2020). To address challenges arising from composite hypotheses and the instability of parameter estimation caused by shrinking active sets, we further propose the Oracle Composite (OC) and Data-driven Composite (DC) tests. These hybrid procedures combine stagewise elimination with intersection-based testing, ensuring reliable estimation and improved practical applicability. Simulation studies demonstrate that the proposed methods substantially reduce the total sample size relative to existing sequential procedures while maintaining rigorous error control. Overall, this thesis advances the theory of large-scale multiple testing in both fixed-sample and sequential settings by establishing new optimality results under dependence and developing adaptive procedures that simultaneously achieve rigorous error control, finite stopping times, and improved sampling efficiency.
On Universal C∗ Algebras associated to Operator Spaces and Generalized Crossed Products
(2026-07-08) Banik, Sayan Kansa
In my thesis, we study generalized crossed product constructions of the group $ C^* $-algebra $ C^*(G) $ with respect to certain completely positive maps, where $ G $ is assumed to be a discrete amenable group. We also investigate the universal $ C^* $-algebra $ \mathcal{E}_\alpha $ introduced by Hirshberg, which is constructed from $ C^*(G) $ and a pure injective homomorphism $ \alpha \colon G \to G $. In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of $ C^*(G) $ induced by $ \alpha $, together with an appropriate choice of transfer operator. In addition, we study crossed products of $ C^*(G) $ arising from states and conditional expectations of the form $ E_H \colon C^*(G) \to C^*(H) $, where $ H < G $ is a proper subgroup. We examine how generalized crossed product constructions change when passing from endomorphism-based framework to to the case of completely positive maps. We study crossed products of $ C^* $-algebras with respect to states, focusing in particular on $ C^*(G) $ equipped with its canonical trace and construct a spatial representation of the system isomorphic to the universal crossed product construction. Further, we show that when $ G $ is virtually abelian, there exists no spatial representation of the system $ (C^*(G), E_H) $ inside $ B(\ell^2(G)) $. Finally, we construct a specific spatial representation of this system and show that, when $[G:H]=\infty$ , the resulting spatial $ C^* $-algebra is isomorphic to the corresponding universal crossed product In the other half, we make a detailed study of operator spaces associated to Brown's noncommutative unitary $C^\ast$-algebra $\mathcal{U}^{nc}_n$ and related $C^\ast$-algebras. Specifically, we identify the universal $C^{\ast}$-algebra $C^{\ast}\langle M_n(\C)^{\ast}\rangle$ of the operator space $M_n(\C)^{\ast}$ with the non-commutative $C^{\ast}$-algebra $\Y^{nc}_n$, the universal unital $C^{\ast}$-algebra generated by elements $u_{ij}$, $1\leq i,j\leq n$ satisfying the relations which make $[u_{ij}]$ a contractive matrix. We also exhibited several operator algebraic properties of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$- in particular, we study the Lifting property (LP), residual finite dimensionality and primitivity of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$. Further, we study the maximal and minimal operator space structure of the standard generators of $\U^{nc}_n$ as well as $\U^{nc}_{n, red}$. Finally, we discuss a natural compact quantum semigroup structure on $C^{\ast}\langle M_n(\C)^{\ast}\rangle$, characterizing invertible elements in its state space under convolution.
Consumer Welfare and Privatization in Mixed Markets: A Developing Country Perspective
(2026-07-14) Dutta, Priyanka
This thesis is presented in four chapters, each examining a distinct aspect of mixed market structures, with a consistent focus on consumer surplus as the primary metric for evaluating the desirability of privatization or private provision in the context of developing countries. The first chapter introduces a symmetric Cournot oligopoly model in which both public and private firms produce a homogeneous good and compete in quantities. We find that consumer surplus is maximized at the two extremes: when the market consists solely of public firms or solely of private firms. In contrast, mixed regimes consistently yield intermediate outcomes, never achieving either the highest or the lowest level of consumer surplus. This result is robust to the level of competition, the specific objective functions assigned to public firms, and holds across all log-concave demand functions and convex cost functions. Importantly, when cost functions are strictly convex, we show that, contrary to conventional wisdom, an increase in the number of firms does not necessarily support privatization; in fact, it may weaken the case for introducing private firms. The first chapter further extends the analysis to Bertrand competition with differentiated products, where firms compete in prices rather than quantities. Unlike in Cournot settings, firms' strategies are strategic complements under Bertrand competition. Despite this fundamental difference in the nature of competition, the key finding persists: mixed oligopolies never maximize consumer surplus. Moreover, in some cases, mixed markets can yield the lowest consumer surplus, even compared to fully public or private regimes. This counterintuitive outcome stems from the regime-contingent behavior of public firms. That is, an inefficient public firm with welfare concerns may respond to rival pricing by setting a higher price in a mixed regime than it would in a fully public one, thereby dampening consumer surplus. While the first chapter shows that mixed markets never yield the highest or lowest consumer surplus, this finding appears at odds with their widespread existence and institutional support across the globe. Several explanations may account for this disconnect. First, privatization decisions are often driven by objectives such as profitability or broader welfare considerations, rather than consumer surplus alone. Second, governments concerned with consumer surplus may still prefer a mixed regime in settings with weak competition policy, where full privatization could increase the risk of collusion. However, in the next two chapters, we demonstrate that mixed regimes can, in fact, top the consumer surplus ranking without resorting to alternative welfare metrics or relying on collusion-based explanations. What is required is to move beyond the symmetric, single-stage oligopoly framework used in the first chapter. The second chapter relaxes the assumption of symmetric firms by introducing cost heterogeneity, a key real-world feature, as firms often differ in cost structures for reasons unrelated to ownership. While privatization can improve efficiency, it may not fully eliminate these underlying cost differences. In this setting, we show that mixed regimes can deliver the highest consumer surplus. For instance, in a duopoly, privatizing the inefficient firm while retaining public ownership of the efficient one can outperform both fully public and fully private regimes. Conversely, if privatization targets the more efficient firm, leaving the less efficient one public, consumer surplus can be the lowest among all ownership structures. These results suggest that ownership design should account for firm-specific efficiency differences, and that optimal privatization policy may be highly context-dependent. The third chapter addresses another limitation of earlier models by considering a vertically related market, with upstream and downstream monopolists interacting in a two stage production process. Public firms in both sectors introduce two layers of inefficiency, while private firms in both sectors generate two rounds of markups, a classic double marginalization problem. We show that a mixed regime can yield the highest consumer surplus by eliminating one layer of inefficiency and one round of markup. This outcome is most likely when markups are moderate and inefficiencies are unevenly distributed across the two sectors. If the inefficiency of public firms is symmetric across the upstream and downstream sectors, mixed regimes tend to perform intermediately or even poorly in terms of consumer surplus. However, when public firm inefficiencies differ sufficiently across sectors, and markup levels are not too high, a mixed structure can outperform both extremes. Extending the model to a richer setting with both upstream and downstream oligopoly shows that the desirability of mixed regimes persists and, in fact, becomes stronger as competition intensifies. Thus, the interaction between vertical structure, firm efficiency, and market power plays a critical role in shaping welfare outcomes in privatization decisions. While the first three chapters evaluate privatization using consumer surplus as a welfare metric under standard oligopoly assumptions, the fourth chapter introduces a new dimension: congestion. In many markets such as healthcare, education, telecommunications, transportation etc., congestion disutility arises as firms serve more consumers, reducing individual utility. We model a Cournot oligopoly with congestion under both mixed and fully private regimes and characterize equilibrium outcomes in the presence of congestion. We conduct comparative static analysis with respect to market size and competition and determine a consumer surplus threshold: a fully private regime yields a higher consumer surplus if the relative cost inefficiency of the public firm exceeds this threshold. We show that congestion and increases in market size both lower this threshold, making privatization more favorable. However, the effect of competition is more nuanced. An increase in competition facilitates privatization only when the initial level of competition is low. Beyond a certain point, additional competition could in fact facilitate public provision. These results highlight how market structures and conditions influence optimal ownership structure in the presence of congestion. Collectively, the four chapters of this thesis underscore the importance of evaluating privatization and ownership design through the lens of consumer welfare, particularly in the context of developing economies. The results challenge simplistic assumptions about public versus private provisions and offer a nuanced framework for understanding when mixed markets can be not just a compromise, but an optimal institutional structure.
Design of reliability acceptance sampling plans
(Indian Statistical Institute, 2026-07-27) Das, Rathin
A reliability acceptance sampling plan (RASP) is used for sampling and decision-making in the acceptance or rejection of a lot of products based on lifetime data obtained from a life test. In practice, censored life tests are employed due to limitations in cost, time, and other testing resources for the collection of lifetime data. This thesis develops the design of optimal RASPs under various censoring schemes and testing environments. Design of optimal Bayesian RASPs (BRASPs) are considered under interval censoring schemes (ICS) and hybrid censoring schemes using Bayesian decision-theoretic approaches. These models incorporate the adversarial relationship between manufacturers and consumers, who differ in prior beliefs and utility functions. Global market competitiveness and rapid technological advancement have pushed manufacturers to produce products with very high reliability. For such products, the mean time to failure under normal operating conditions is often prohibitively long. To address this issue, a BRASP based on a novel adaptive simple step-stress partial accelerated life test (ASSSPALT) framework is proposed under Type-I censoring using common prior and utility functions. The adaptive scheme dynamically adjusts stress levels based on observed failures, providing a general framework that accommodates both accelerated and non-accelerated testing under Type-I censoring. For complex products, failure may occur due to multiple causes. The work considers the design of RASP for competing risk data under progressive Type-I interval censoring using the producer's and consumer's risk approaches. The asymptotic properties of maximum likelihood estimators are derived to develop optimal plans. A frailty-based model is employed to capture dependence among competing risks and evaluate its influence on sampling plan performance. Subsequently, RASP is extended to a Bayesian framework under interval censoring. Further, for complex products with very high reliability, the design of BRASP is considered based on ASSSPALT under Type-II censoring for competing risk data. This framework unifies both accelerated and non-accelerated testing scenarios for competing risk data under Type-II censoring. The proposed methodologies for designing RASPs are illustrated using real-life data.
Distributed Computation of Graph Structures by Mobile Agents
(2026-07-15) Chand, Prabhat Kumar
This thesis investigates how mobile agents with no centralised control can be employed in anonymous networks to perform efficient distributed graph computations. The network is modelled as a simple, undirected, anonymous graph with n nodes and m edges, where nodes are memoryless and indistinguishable, and edges represent bidirectional communication links or traversal paths for the agents. The mobile agents are uniquely identifiable, possess limited local memory, and operate under a local communication model, in which communication is restricted to agents colocated at the same node. Under this computational model, we explore how mobile agents can collaborate effectively to solve global network problems—including dispersion in the presence of crashes, the construction of spanning trees, the identification of dominating sets, and the analysis of sub-graph hierarchies. We study the time complexity and memory usage per agent required to solve the above problems. The first contributory chapter addresses the fault-tolerant dispersion problem, aiming to evenly spread mobile agents across an anonymous graph in the presence of crash faults, from two initial configurations: rooted, where all the agents start at a single node, and arbitrary, where agents are initially scattered across the graph in multiple clusters. This chapter explores how dispersion can be achieved under both settings, ensuring that each node eventually hosts at most one functional agent despite crashes. The next chapter presents the problem of computing dominating sets using mobile agents, where two different algorithms are introduced: one for computing minimal dominating sets in O(m) time when the agents are gathered at a single node, and another for scenarios where agents start from multiple clusters. Additionally, an ln ∆ approximation algorithm for the minimum dominating set problem is provided, where ∆ is the maximum degree of the graph. The subsequent chapter focuses on subgraph analytics using mobile agents dispersed across the nodes of a graph. We present algorithms for triangle counting (3-cycles) in general graphs and butterfly counting (4-cycles) in bipartite graphs. The triangle counting framework extends to related problems such as truss decomposition, triangle centrality, and local clustering coefficient. These methods enable the distributed identification of cohesive structures and dense subgraphs, with butterfly counting being of relevance to bipartite graphs commonly found in social network analysis and recommendation systems. The final chapter focuses on constructing tree structures, specifically BFS trees and minimum spanning trees, using mobile agents. These algorithms, which assume minimal prior knowledge, improve upon existing methods by achieving better time complexity and optimal memory usage. Throughout the thesis, these graph problems are explored through the lens of the mobile agent framework, focusing on minimising the time complexity of the algorithms and memory usage per agent.
A Regression Tree Framework for Denoising and Monitoring of Image Data
(Indian Statistical Institute, 2026-07-02) Basak, Subhasish
The proliferation of advanced image acquisition technologies has led to the routine collection of large-scale image data across numerous scientific domains. This widespread reliance on image data accentuates the imperative to develop robust and efficient imaging techniques, which are essential for supporting modern applications across various scientific and industrial domains. This dissertation focuses on the development and analysis of methods for image denoising and image monitoring, two fundamental tasks in modern image analysis. A wide array of image denoising techniques exists in the literature, each tailored to handle specific types of noise or structural characteristics. However, no single method proves universally optimal, as each comes with its own advantages and trade-offs. In the first part of the dissertation, different configurations of local neighbourhoods are investigated, and an adaptive framework is proposed that combines these with local clustering-based smoothing to effectively harness the advantages of both methodologies. The dissertation then introduces a regression tree-based framework utilizing Oblique-axis Regression Trees (ORT) to estimate discontinuous regression functions in finite-dimensional spaces and applies this methodology to achieve effective image denoising. Due to an alternative set of assumptions on the underlying regression function, the overall structure of the proofs is substantially simpler than those typically found in the existing literature on regression trees. Finally, leveraging the ORT framework, the dissertation introduces an original approach to monitor drift patterns within an image sequence. Even though gradual temporal variations, known as drifts, are frequently observed in image sequences, drift monitoring remains an underexplored research area. This dissertation thus makes an effort to address that gap. Theoretical analysis and numerical studies, conducted on both simulated and real-world data, demonstrate the broad applicability and effectiveness of the proposed methods.
Essays on Liberalization and Trade Policy in Developing Economies under Increasing Returns
(Indian Statistical Institute, 2026-06-25) Kapoor, Shivam
Developing countries occupy an important place in the world trade pattern of today. Taking advantage of low wages in these countries, the low technology and labor intensive stages of production in many product lines are outsourced here. Hence participation in Global Value Chains for a developing country on the one hand generates employment in manufacturing thereby helping realize economies of scale arising out of division of labor, on the other hand the gains from trade accruing from such participation may be low due to presence of severe distortions like weak legal and financial institutions, and may cause distributional conflicts. My dissertation studies the role of trade and investment liberalization policies for developing countries in the presence of increasing returns. Chapter 2 of the thesis focuses on trade in both final good and the differentiated varieties of an intermediate input between a capital abundant country and a labor abundant country. The intermediate input sector comprises monopolistically competitive firms of heterogeneous productivity. I analyze the impact on the two countries of moving from autarky to free trade. When trade in varieties is subject to both fixed and variable trade costs, the impact of a bilateral liberalization is studied with the help of a numerical simulation. Chapter 3 considers a small open economy characterized by open urban unemployment and rural-urban migration. The urban sector produces an import-competing (tariff protected) final good and a non-traded input that is subject to increasing returns to scale. The rural sector produces the exportable by combining the input with rural labor. In this structure the policy impacts of a foreign capital inflow and an increase in tariff protection are studied. In Chapter 4, I consider a two country, three sector and one factor model of trade and unemployment. Trade takes place in one homogeneous good (costlessly) and the varieties of two differentiated industrial goods (subject to variable trade cost). Equilibrium unemployment of the Shapiro-Stiglitz type is modeled. In this structure I analyze the impact of a unilateral, sector-specific trade policy on industrial relocation in both the protected and unprotected sectors, and on the employment rates in the two countries. Turning again to the case of a small open economy, in Chapter 5, I combine the production structure studied in Chapter 2 with the assumption that the Home economy imports a fixed number of Foreign varieties at a given price (subject to a tariff). Here I study the productivity and welfare effects of an inflow of foreign capital assuming full repatriation of foreign profits. Summary, conclusions and possible extensions for future research are outlined in Chapter 6.
Relay Selection and User Scheduling in Reconfigurable Intelligent Surface Assisted Millimeter-wave D2D Communication
(Indian Statistical Institute, 2026-06-19) Sau, Lakshmikanta
Reconfigurable intelligent surface (RIS) assisted millimeter wave (mmWave) device to device (D2D) communication has recently been proposed as a viable solution to support the overwhelming data traffic in fifth-generation (5G) and beyond wireless networks. However, due to the substantial propagation and penetration losses of mmWave, a direct line of sight (LoS) link between a pair of proximity devices is required for effective communication. Static obstacles like trees and buildings can easily obstruct the direct LoS connectivity between a device pair. In such cases, RISs help to establish an indirect LoS link between an obstructed device pair by reflecting the signals in the desired direction. In Chapter 2, we propose a set cover-based RIS deployment strategy to serve the maximum number of obstructed device pairs with the minimum number of RISs. In particular, we have demonstrated that double reflections via two consecutive RISs can greatly lower the RIS density in the environment, preventing resource waste and enabling the service of more obstructed device pairs. After the RIS deployment, for information transfer, we also propose an energy-efficient RIS group selection criteria. Moreover, we prove that under some conditions, double reflections are more beneficial than single reflection, which is counter-intuitive. Numerical results show that our approach outperforms a random and a recent deployment strategy. In Chapter 3, assuming that the RISs are already deployed and the positions of the nearby users are known, we propose a double-RIS assisted multihop routing scheme for a device pair. Besides the RISs, the emphasis of this work is to make more use of the existing intermediate users (IUs), which can act as relays. Hence, the density of RIS deployment in the surroundings can be reduced, which leads to the avoidance of resource wastage. However, we cannot solely depend on the IUs because this implies complete dependence on their availability for relaying and as a result, the aspect of reliability in terms of delay-constrained information transfer may not be guaranteed. Moreover, the IUs are considered capable of energy harvesting, and as a result, they do not waste their own energy in the process of volunteering to act as a relay for other users. Numerical results demonstrate the advantage of the proposed scheme over some existing approaches, and lastly, useful insights related to the scheme design are also drawn, where we characterize the maximum acceptable delay at each hop under different set-ups. The previous chapter presented a routing scheme for a particular device pair. For multiple device pairs, a single RIS may be simultaneously requested by several devices to act as a relay for their seamless communications. To deal with such cases, in Chapter 4, we propose a priority-aware, user traffic–dependent, grouping-based multihop user scheduling scheme for RIS-assisted mmWave D2D communication under spatially correlated channels. Specifically, the proposed scheme exploits the priority of the users (based on their respective delay-constrained applications) and the aspect of spatial correlation in the narrowly spaced reflecting elements of the RISs. Here, based on the other users in the neighborhood, their respective traffic characteristics, and the already deployed RISs in the surroundings, we establish a multihop connection for energy-efficient information transfer from one of the users to its intended receiver. In this context, we take into account the impact of considering practical discrete phase shifts at the RIS patches instead of its ideal continuous counterpart. Moreover, we also claim and demonstrate that the existing classic least remaining distance based approach is not always the optimal solution. Finally, the numerical results demonstrate the advantages of the proposed strategy over the existing benchmark schemes in terms of data throughput, energy consumption, and energy efficiency. In the previous chapter, each device pair selects the best energy-efficient group, and in case of multiple pairs selecting the same best group, they are scheduled as per their priorities. However, the best group may lead to a large delay for some of the pairs. Moreover, the energy harvesting aspect is not considered there. In Chapter 5, we investigate various group selection strategies which select the k-th best group by considering self-sustainable RIS with spatially correlated channels. Specifically, we consider both power splitting and time switching configurations of the self-sustainable RIS to analyze the system performance and propose appropriate bounds on the choice of system parameters. The analysis takes into account both linear and non-linear energy harvesting models. Based on the application requirements, we propose various group selection strategies, which schedule the k-th best available group based on the end-to-end signal-to-noise ratio and also the energy harvested at a particular group. Accordingly, by using tools from high-order statistics, we derive analytical expressions for the outage probability of each selection strategy. Moreover, using extreme value theory, we investigate an asymptotic scenario where the number of groups available for selection at an RIS approaches infinity. The nontrivial insights obtained from this approach are especially beneficial in applications like large intelligent surface-aided wireless communication. Finally, the numerical results demonstrate the importance and benefits of the proposed approaches in terms of throughput and outage performance.
Designing Truthful Mechanisms: A study of Voting and Matching Rules
(Indian Statistical Institute, 2026-05-27) Bose, Abhigyan
Mechanisms are formal procedures used in game theory and economics to aggregate individual preferences into collective outcomes in strategic settings with private information. A truthful or strategy-proof mechanism ensures that agents are best off reporting their true preferences, thereby eliminating incentives for manipulation. Such mechanisms are important for achieving fairness, efficiency, and simplicity in practical applications. In environments without monetary transfers, voting and matching rules are two central classes of mechanisms. Voting rules are typically used for public decision-making, while matching rules allocate private goods among individuals. This thesis studies strategy-proof voting and matching rules under different environments and restrictions on preference domains. The thesis is divided into two parts: the first three chapters focus on voting rules, and the last three on matching rules. The first two chapters analyze voting environments where agents share common beliefs over the preference domain, with the relevant incentive notion being locally robust ordinal Bayesian incentive compatibility (LOBIC). Chapter one corrects a result from the existing literature, while chapter two introduces a model of correlated priors based on a betweenness property. Chapter three studies voting structures in environments where agents are connected through an exogenous graph. In this setting, we introduce incentive-compatible extendable voting rules and characterize them under different graph structures. The remaining chapters focus on matching theory. Under unrestricted preferences, the Top Trading Cycles (TTC) mechanism of Gale is known to be the unique rule satisfying strategy-proofness, individual rationality, and Pareto optimality. However, Bade showed that this uniqueness fails under single peaked preferences. Chapters four and five extend this line of research to broader preference domains: multiple single-peaked preferences and single peaked preferences on trees. In both chapters, we propose new algorithms satisfying strategy-proofness, individual rationality, and Pareto optimality. We further study the stronger notion of obvious strategy-proofness, introduced by Li (2017), proving non-existence results for such matching rules. Finally, chapter six presents general existence results for obviously strategy-proof matching rules within the class of individually rational and Pareto optimal rules.
C*-Extreme Quantum Instruments; Completion and Disintegration of Completely Positive Maps
(2026-06-01) Chongdar, Arghya
This thesis develops a comprehensive operator-algebraic framework for the study of completely positive (CP) instruments, with contributions spanning convexity theory, integration theory, completion problems, and disintegration theory. We begin by laying the foundational groundwork in the theory of C*-algebras and von Neumann algebras, introducing key structures such as CP maps, positive operator-valued measures (POVMs), and CP instruments, along with their dilation-theoretic properties. Pure and decomposable instruments are characterized via minimal bi-dilations, and CP instruments are realized as bivariate maps, providing a rigorous quantum analogue of classical joint measures. The thesis then investigates convexity-theoretic aspects of instruments. A structural characterization of C*-extreme unital completely positive (UCP) instruments on finite-dimensional Hilbert spaces is established, employing methods from the theory of nest algebras. The interplay between C*-extremality and the marginals of an instrument is studied, yielding results on the spectral nature of POVM marginals and the unique determination of an instrument from a single C*-extreme marginal. A systematic integration theory with respect to CP instruments is then developed, inspired by Bartle's classical vector integration framework. This culminates in a CP-instrument correspondence theorem on compact Hausdorff spaces and, notably, a Krein-Milman type theorem for CP instruments on separable C*-algebras — a result not previously available in the literature. The thesis further addresses the CP completion problem — the extension of partially defined linear maps to fully CP maps on C*-algebras — establishing the existence and uniqueness of minimal CP completions, and generalizing a result of Parzygnat and Russo on almost-everywhere identity maps to the full generality of von Neumann algebras. Finally, the theory of non-commutative disintegration is developed, connecting classical disintegration to the existence of left inverses for CP maps. Structural results for left-invertible normal CP maps on B(H) are obtained, and existence and uniqueness of disintegrations are established in the infinite-dimensional setting.

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