Probability Theory: Springer Universitext Series

• Universitext is a series of textbooks for master's level and beyond, covering a wide range of mathematical disciplines.
• These books often include informal and personal approaches, evolving through editions to reflect teaching curriculum changes.
• Cutting-edge courses and research topics can find their way into Universitext.

Probability Theory

• The mathematical theory of probabilities originated from defining probability as the ratio of favorable outcomes to the total equally likely outcomes (the "classical" approach).
• R. von Mises criticized this approach as too restrictive, basing his conception on the stability of event frequencies in long experiment series.
• Von Mises' approach was criticized for confusing physical and mathematical concepts and for assuming frequencies would always have a limit.
• S.N. Bernstein outlined general features, and A.N. Kolmogorov provided a complete axiomatic theory in 1933 in his book "Foundations of Probability Theory."
• Kolmogorov's axiomatics is now universally accepted because it removed obstacles and solved Hilbert's sixth problem.

Asymptotics and Distributions

• Focus on asymptotics of probabilities related to random variables χ and S, including P(χ+ > x | η+ < ∞), P(χ−0 < −x), and P(S > x).
• Study the distribution of maximal values of generalized renewal processes.
• Examines the distribution of the first passage time, including properties of the distributions of times η± and the first passage time of an arbitrary level x by arithmetic skip-free walks.

Markov Chains

• Includes definitions, examples, and state classifications.
• Discusses necessary and sufficient conditions for state recurrence, types of states in an irreducible chain, and the structure of a periodic chain.
• Explores symmetric random walks in Rk (k ≥ 2) and arbitrary symmetric random walks on the line.
• Features ergodic theorems, the law of large numbers, and the central limit theorem for the number of visits to a given state.
• Discusses the behavior of transition probabilities for reducible chains.
• Covers Markov chains with arbitrary state spaces and ergodicity of chains with positive atoms.
• Includes the ergodic theorem and conditions (I) and (II).
• Laws of large numbers and the central limit theorem are given for sums of random variables defined on a Markov chain.

Events and Probabilities

• The union (or sum) of events A and B, denoted A ∪ B (or A + B), includes elementary outcomes in either A or B.
• The intersection (or product) of events A and B, denoted AB (or A ∩ B), includes elementary outcomes in both A and B.
• The difference of events A and B, denoted A − B (or A \ B), includes elements in A but not in B.
• The certain event is denoted by Ω, and the impossible event is denoted by ∅.
• The complement of event A is denoted by A = Ω − A.
• Mutually exclusive events A and B satisfy AB = ∅.
• In an experiment of rolling a die twice, the space of elementary events consists of 36 elements (i, j), where i and j are the outcomes of the first and second rolls, respectively.

Probabilities of Elementary Events

• Probabilities of elementary events involve a nonnegative function P on Ω such that the sum of P(ω) over all ω in Ω equals 1.
• P(A), the probability of event A, is the sum of P(ω) for all ω in A.
• For a symmetric die, P(1) = P(2) = ... = P(6) = 1/6; for a symmetric coin, P(h) = P(t) = 1/2.
• An example is the calculation of probability for a coin toss experiment stopping on an even step.

Properties of Probability

• (1) P(∅) = 0, P(Ω) = 1.
• (2) P(A + B) = P(A) + P(B) − P(AB).
• (3) P(A) = 1 − P(A).
• For disjoint events A and B, P(A + B) = P(A) + P(B).
• For an arbitrary number of disjoint events A1, A2,..., P(A1 + A2 + ...) = P(A1) + P(A2) + ....

Conditional Distribution

• The conditional distribution of η (where η ∈ [−π/2, π/2]) occurs with a probability P(η ∈ [−π/2, π/2]) = 1/2 and coincides with U−π/2,π/2.
• With (ξ − α)/σ = tan η, P(ξ < x) = (1/π)arctan((x − α)/σ) + 1/2.

Distributions

• The coordinates of traces on the x-axis of particles emitted from the point (α, σ) follow the Cauchy distribution Kα,σ.
• The Poisson distribution λ assumes nonnegative integer values with probabilities P(ξ = m) = (λ^m / m!) * e^(-λ) for λ > 0, m = 0, 1, 2, ...
• Poisson distribution function: F(x) = sum(λ^m * e^(-λ) / m!) for m >= 0 when x > 0, and 0 for x ≤ 0.

Discrete Distributions

• The distribution of a random variable ξ is discrete if ξ assumes finitely or countably many values x1, x2,... so that pk = P(ξ = xk) > 0, and the sum of pk = 1.
• A discrete distribution {pk} can always be defined on a discrete probability space.
• Discrete distributions can be characterized by a table of values (x1, x2, x3, ...) and probabilities (p1, p2, p3, ...).
• Examples of discrete distributions include Ia, Bnp, λ, and geometric distributions.

Explore Probability Theory within Springer's Universitext series. This text covers the evolution of probability from classical definitions to Kolmogorov's axioms. It also discusses limitations of early approaches and the impact of the 'Foundations of Probability Theory'.