  The article studies the running maxima $Y_{m,j}=\max_{1 \le k \le m, 1 \le n
\le j} X_{k,n} - a_{m,j}$ where $\{X_{k,n}, k \ge 1, n \ge 1\}$ is a double
array of $\varphi$-subgaussian random variables and $\{a_{m,j}, m\ge 1, j\ge
1\}$ is a double array of constants. Asymptotics of the maxima of the double
arrays of positive and negative parts of $\{Y_{m,j}, m \ge 1, j \ge 1\}$ are
studied, when $\{X_{k,n}, k \ge 1, n \ge 1\}$ have suitable "exponential-type"
tail distributions. The main results are specified for various important
particular scenarios and classes of $\varphi$-subgaussian random variables.
