almost sure limit theorem

Target entity: almost sure limit theorem
Context triple: [Khinchin's law of the iterated logarithm, typeOfLimit, almost sure limit theorem]

• A. Borel–Cantelli lemmas
  The Borel–Cantelli lemmas are fundamental results in probability theory that characterize when events occur infinitely often or only finitely often, based on the convergence or divergence of the sum of their probabilities.
• B. Kolmogorov zero–one law
  The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
• C. Limit Laws for Sums of Independent Random Variables
  Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
• D. monotone convergence theorem
  The monotone convergence theorem is a fundamental result in measure theory stating that the integral of a pointwise increasing sequence of nonnegative measurable functions equals the limit of their integrals.
• E. Lyapunov central limit theorem
  The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: almost sure limit theorem
Target entity description: An almost sure limit theorem is a probabilistic result that describes the precise pathwise asymptotic behavior of random variables, guaranteeing convergence with probability one rather than just in distribution or in expectation.

• A. Borel–Cantelli lemmas
  The Borel–Cantelli lemmas are fundamental results in probability theory that characterize when events occur infinitely often or only finitely often, based on the convergence or divergence of the sum of their probabilities.
• B. Kolmogorov zero–one law
  The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
• C. Limit Laws for Sums of Independent Random Variables
  Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
• D. monotone convergence theorem
  The monotone convergence theorem is a fundamental result in measure theory stating that the integral of a pointwise increasing sequence of nonnegative measurable functions equals the limit of their integrals.
• E. Lyapunov central limit theorem
  The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
• F. None of above. chosen