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$\Rightarrow$ is convergence of random processes by finite-dimensional distribution.

$$P(\overline{lim_{n \to \infty}} \frac{S_n}{\sqrt{2n\log\log n}} = 1) = 1$$

$$P(\underline{lim_{n \to \infty}} \frac{S_n}{\sqrt{2n\log\log n}} = -1) = 1$$

$$P(max_{1 \leq k \leq n} S_k \geq t) \leq \frac{n Var(X_1)}{t^2}$$

Wiener Theorem

If $E(X_1) = 0$ and $0 < Var(X_1) < +\infty$, then

where, $W(t)$ stands for Wiener process.

Hopf lemma

$$P(S_n > 0) \geq 1 - \frac{P(X_1 = 0)}{nP(X_1 > 0)}$$

$\Rightarrow$ is convergence of stochastic processes in Skorokhod space.

$$P(\overline{\lim_{n \to \infty}} \frac{S_n}{\sqrt{2n\log\log n}} = 1) = 1$$

$$P(\underline{\lim_{n \to \infty}} \frac{S_n}{\sqrt{2n\log\log n}} = -1) = 1$$

$$P(\max_{1 \leq k \leq n} S_k \geq t) \leq \frac{n Var(X_1)}{t^2}$$

Donsker invariance principle

If $E(X_1) = 0$, $0 < Var(X_1) < +\infty$ and $t \in [0; 1]$ then

where, $W(t)$ stands for Brownian motion.

Liggett invariance principle

If $E(X_1) = 0$, $0 < Var(X_1) < +\infty$, $t \in [0; 1]$ and $a > 0$ then

$$\frac{S_{\lfloor nt \rfloor}}{\sqrt{n Var(X_1)}} | S_n \in (-a; a] \Rightarrow B(t)$$

where, $B(t)$ stands for Brownian bridge.

Eagleheart invariance principle

If $E(X_1) = 0$, $0 < Var(X_1) < +\infty$ and $t \in [0; 1]$ then

$$\frac{S_{\lfloor nt \rfloor}}{\sqrt{n Var(X_1)}} | \min\{n > 0| S_n \leq 0\} > n \Rightarrow W^+(t)$$

where, $W^+(t)$ stands for Brownian meander.

Hopf lemma

$$P(S_n > 0) \geq \frac{nP(X_1 = 0)}{1 + (n-1)P(X_1 = 0)}$$

If you already know all these facts and want something more exotic, then sorry (however, if I find anything else, I will expand this list)

Cramer Theorem

If $\forall t \in \mathbb{R}$ $E[e^{tX_1}]<+\infty$, then

$$\lim_{n \to \infty} \frac{1}{n}\ln(P(S_n \geq nx)) = \inf_{t \in \mathbb{R}}(\ln(E[e^{tX_1}])-tx)$$

If you already know all these facts and want something more exotic, then sorry (however, if I find anything else, I will expand this list)

Suppose $P(X_1 \geq 0) = 1$, then

If $E(X_1) = 0$ and

Suppose $P(X_1 \geq 0) = 1$ and $P(X_1 > 0) > 0$, then

Suppose $E(X_1) = 0$ and