This article introduces an analytical framework that interprets individual measures of entropy-based mobility derived from mobile phone data. We explore and analyze two widely recognized entropy metrics: random entropy and uncorrelated Shannon entropy. These metrics are estimated through collective variables of human mobility, including movement trends and population density. By employing a collisional model, we establish statistical relationships between entropy measures and mobility variables. Furthermore, our research addresses three primary objectives: firstly, validating the model; secondly, exploring correlations between aggregated mobility and entropy measures in comparison to five economic indicators; and finally, demonstrating the utility of entropy measures. Specifically, we provide an effective population density estimate that offers a more realistic understanding of social interactions. This estimation takes into account both movement regularities and intensity, utilizing real-time data analysis conducted during the peak period of the Covid-19 pandemic.

We investigate the impact of aging on exchangeable inter-arrival times in mixed renewal processes, exploring its implications for reliability and survival analysis. In this study, first, we revisit the definition of renewal point processes, where inter-event time intervals are considered as exchangeable non-negative random variables. Then, we define the concept of statistical aging as latency in the observational process of event counting. Latency affects event detection but preserves exchangeability. However, it may alter the statistical properties of inter-event time intervals. Our analytical and numerical assessments highlight the significance of aging in exchangeable lifetimes, offering insights into key metrics such as the failure survival function, renewal function, and hazard rate function. Through archetypal examples and empirical findings, we illustrate the implications of aging on renewal processes. In particular, employing a Bayesian perspective, we analyze high-frequency currency exchange rate data to assess the impact of aging on volatility risk evaluation. This study contributes novel insights to the literature of renewal theory and survival analysis, emphasizing the role of latency aging in stochastic point processes under exchangeability assumption.

This paper explores the application of Random Matrix Theory (RMT) as a methodological enhancement for portfolio selection within financial markets. Traditional approaches to portfolio optimization often rely on historical estimates of correlation matrices, particularly susceptible to instability. To address this challenge, we combine a data preprocessing technique based on the Hilbert transformation of returns with RMT to refine the accuracy and robustness of correlation matrix estimation. By contrasting empirical correlations with those derived from random matrices, we uncover non-random properties and underlying relationships within financial data. We then utilize this methodology to construct the correlation network dependence structure used in portfolio optimization. The empirical analysis presented in this paper validates the effectiveness of RMT in enhancing portfolio diversification and risk management strategies. This research contributes by offering investors and portfolio managers with methodological insights to construct portfolios that are more stable, robust, and diversified. At the same time, it advances our comprehension of the intricate statistical principles underlying multivariate financial data.