Stochastic processes. List of topics covered in the course.
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Liminf and limsup for random events.
Borel-Cantelli lemmas.
Tail sigma-algebra and Kolmogorov's zero-one law.
Inequalities: Jensen, Markov, Chebyshev, Cauchy-Schwartz, Hoelder, Lyapunov, Minkowski, Kolmogorov.
Modes of convergence: almost sure, in mean, in L^r, in probability, in distribution.
Implications and partial rverse implications for different modes of convergence. Counterexamples.
Continuity of expectations: bounded convergence, Fatou lemma, monotone convergence, dominated convergence.
Uniform integrability and convergence in mean.
Weakly and strongly stationary processes. Autocorrelation function.
Best linear predictor.
AR(1) process.
Linear combinations of sinusoids.
Spectral distribution function and spectral density.
Spectral representation of weakly stationary processes with continuous and discrete time (without proof).
Stochastic integrals (basic idea).
Ergodic theorem for the weakly stationary processes (proof sketch).
Ergodic theorem for the strongly stationary processes (without proof). Examples and counterexamples.
Stationary Gaussian processes. Ornstein-Uhlenbeck process.
Renewal process. Renewal function. Renewal equation.
Excess life and current lifetime.
Poisson process, waiting time paradox.
Law of large numbers and Central limit theorem for the renewal processes.
Stopping times. Wald's equation.
Elementary renewal theorem.
Renewal theorem (without a proof). Key renewal theorem (proof sketch).
Delayed renewal process.Stationary case.
Renewal-reward process. Renewal-reward theorem.
Regeneration technique for queues. Little's law.
Annotations system for queueing systems.
Traffic intensity parameter.
M/M/1 queues. Kolmogorov forward equations.
Stationary regime for M/M/1 queues.
M/G/1 queues. Busy period distribution. Embedded branching process.
Stationary regime for M/G/1 queues.
G/M/1 queues. The embedded A-chain. The waiting time at equilibrium.
G/G/1 queues. Lindley equation. Wiener-Hopf equation.
Embedded random walk and ladder points. The waiting time at equilibrium.
Martingales, submartingales, supermartingales.
Examples. De Moivre martingale.
Doob-Kolmogorov inequality. L^2 convergence of martingales.
Application to branching processes.
Predictability. Doob decomposition.
Martingale properties of transformed martingales.
Examples: optional stopping and optional starting.
Hoeffding inequalty. Application to the large deviations.
Snell uppcrossing inequality. L^1 convergence theorem for submartingales.
Doob martingale.
Bounded stopping times. Optional sampling theorem.
Unbounded stopping times. Optional stopping theorems.
Wald identity.
Maximal inequalities.
Convergence in L^r for martingales.
Backward martingales. Strong law of large numbers.