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Root number
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7130 |
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Semester
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HS2024 |
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Type of course
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Lecture |
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Allocation to subject
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Statistics |
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Type of exam
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not defined |
| Title |
Advanced Course in Probability |
| Description |
Format: Presence lectures and tutorials
Sigma-algebra and probability measure. Continuity properties of probability measure.
Borel sigma-algebra. Measurability of continuous functions.
Random element. Convergence almost surely. Approximation by simple and elementary random elements. Skorokhod theorem (without proof).
Regular measures.
Tight measures and Ulam's theorem.
Support of a measure (defintions).
Weak convergence. Portmanteau theorem. Convergence determining classes. Special cases of convergence on finite and countable spaces. Special case of weak convergence of random variables. Continuous mapping theorem.
Convergence almost surely and in probability for random elements. The Skorokhod-Dudley theorem (single probability space method).
Wichura's theorem and its application to continuous functions.
Uniform integrability.
The Prokhorov metric and the total variation metric.
The Prokhorov theorem (Urysohn's lemma without proof).
The Lindeberg theorem.
Characteristic functions. Uniqueness and convergence.
Cram\'er-Wold device.
Scheffe's theorem (convergence of densities implies the convergence in the total variation metric).
Infinitely divisible distributions. Properties of their characteristic functions.
Stable distributions (examples: normal, Cauchy). Khinchin's lemma.
Weak convergence of random continuous functions (formulation, relationship to Wichura's theorem). Tightness condition using moments (without proof).
Invariance principle (formulation, main steps of the proof - step functions, bounds for increments using the Kolmogorov's inequality).
Zero-one laws of Kolmogorov (formulation) and Hewitt-Savage (proof). Lemma of Borel--Cantelli (formulation and relationship to the zero-one law).
Convergence of series (sufficient conditions for the convergence with proofs). The Kolmogorov inequalities (direct inequality with proof).
Strong laws of large numbers (Lemmas of Toeplitz and Kroenecker without proof).
Law of the iterated logarithm (formulation of the Hartman-Wintner theorem, proof only for the case of normal summands).
Large deviations (use of the Markov inequality,
proof of the Cram\'er-Chernoff theorem only in case of finite moment
generating function). |
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ILIAS-Link (Learning resource for course)
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Registrations are transmitted from CTS to (no admission in ILIAS possible).
ILIAS
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Link to another web site
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| Lecturers |
Prof. Dr.
Ilya Molchanov, Institute of Mathematical Statistics and Actuarial Science ✉
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ECTS
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6 |
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Recognition as optional course possible
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Yes |
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Grading
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1 to 6 |
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| Dates |
Monday 13:15-15:00 Weekly
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Thursday 13:15-15:00 Weekly
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Rooms
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| Students please consult the detailed view for complete information on dates, rooms and planned podcasts. |
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