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Neka je \(\{X_n : n \in \mathbb{N}\}\) niz nezavisnih i jednako distribuiranih slučajnih varijabli. Slučajna šetnja definirana je sa: \(S_0 = 0\) i \(S_n = X_1+ \ldots +X_n\) za \(n \in \mathbb{N}\). Neka je \(N = inf\{n \in \mathbb{N} : S_n > 0\}\) prvo vrijeme posjeta skupu \( (0; \infty)\) te \(\bar{N} = inf\{n \in \mathbb{N} : S_n \leq 0\}\) prvo vrijeme posjeta skupu \( (-\infty; 0] \). U teoremu 3.2.3. dokazana je sljedeća tvrdnja. Ako je \(P\{X_1 = 0\} < 1\) tada vrijedi \( S_n \xrightarrow{n \to \infty} \infty \) (g.s.) ili \( S_n \xrightarrow{n \to \infty} -\infty \) (g.s.) ili \( -\infty = \underset{n \to \infty}{\liminf} S_n < \underset{n \to \infty}{\limsup} S_n = \infty \) (g.s.). Pomoću Wiener-Hopfove faktorizacije (teorem 4.1.7.) dokazane su Baxterove jednakosti (teorem 4.2.2.) i Spitzerova formula (teorem 4.3.1.). Baxterove jednakosti određuju distribuciju slučajnih vektora \( (N; S_N) \) i \( (\bar{N}, S_{\bar{N}}) \). Spitzerova formula određuje distribuciju procesa maksimuma \( \{ max_{0 \leq k \leq n} S_k : n \in \mathbb{N}_0 \} \). Let \(\{X_n : n \in \mathbb{N}\}\) be a sequence of independent and identically distributed random variables. Random walk is defined by \(S_0 = 0\) and \(S_n = X_1+ \ldots +X_n\) for each \(n \in \mathbb{N}\). Let \(N = inf\{n \in \mathbb{N} : S_n > 0\}\) be first visit to the set \((0; \infty)\), and let \(\bar{N} = inf\{n \in \mathbb{N} : S_n \leq 0\}\) be first visit to the set \((-\infty; 0]\). Theorem 3.2.3 proves next statement. If \(P\{X_1 = 0\} < 1\) then \(S_n \xrightarrow{n \to \infty} \infty\) (g.s.) or \(S_n \xrightarrow{n \to \infty} -\infty\) (g.s.) or \(-\infty = \underset{n \to \infty}{\liminf} S_n < \underset{n \to \infty}{\limsup} S_n = \infty\) (g.s.) Wiener-Hopf factorization (Theorem 4.1.7.) proved Baxter's equations (theorem 4.2.2) and Spitzer's formula (Theorem 4.3.1.). Baxter's equations determines distribution of random vectors \((N; S_N)\) and \((\bar{N}, S_{\bar{N}})\). Spitzer's formula determines distribution of the maximum process \( \{ max_{0 \leq k \leq n} S_k : n \in \mathbb{N}_0 \}\). |
